Construction of the Noncommutative Rank I Bergman Domain

Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 36)


In this paper we present a harmonic oscillator realization of the most degenerate discrete series representations of the SU(2,1) group and the deformation quantization of the coset space \(D = SU(2,1)/U(2)\) with the method of coherent state quantization. This short article is based on a talk given at the 9-th International Workshop, Varna “Lie Theory and Its Applications in Physics” (LT-9).


Coherent State Star Product Discrete Series Bergman Kernel Coset Space 
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.



The author is very grateful to Harald Grosse and Peter Presnajder for useful discussions.


  1. 1.
    Bieliavsky, P., Gurau, R., Rivasseau, V.: Non commutative field theory on rank one symmetric spaces. J. Noncommut. Geom. 3, 99–123 (2009). [arXiv:0806.4255 [hep-th]]Google Scholar
  2. 2.
    Bortwick, D., Lesniewski, A., Upmeier, H.: Non-perturbative deformation quantization of Cartan domains. J. Funct. Anal. 113, 153–176 (1993); Upmeier, H.: Toeplitz operators and index theory in several complex variavbles, Harmonic Analysis 1996. Birkhäuser, Basel (1996)Google Scholar
  3. 3.
    Connes, A.: Noncommutative Geometry. Academic Press, Boston (1994)MATHGoogle Scholar
  4. 4.
    Douglas, M.R., Nekrasov, N.A.: Noncommutative field theory. Rev. Mod. Phys. 73, 977 (2001) [hep-th/0106048]MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Grosse, H., Prešnajder, P.: The construction of noncommutative manifolds using coherent states. Let. Math. Phys. 28, 239–250 (1993)MATHCrossRefGoogle Scholar
  6. 6.
    Grosse, H., Wulkenhaar, R.: Renormalisation of phi**4 theory on noncommutative R**4 in the matrix base. Commun. Math. Phys. 256, 305 (2005) [arXiv:hep-th/0401128]MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Grosse, H., Presnajder, P., Wang, Z.: Quantum field theory on quantized bergman domain. J. Math. Phys. 53, 013508 (2012) [arXiv:1005.5723 [math-ph]]Google Scholar
  8. 8.
    Jakimowicz, G., Odziewicz, A.: Quantum complex Minkowski space. J. Geom. Phys. 56, 1576–1599 (2006)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Kirillov, A.A.: Elements of Theory of Representations. Springer, Berlin (1976); Representation theory and noncommutative harmonic analysis II. Encyklopedia Math. Sci. 59, Springer, Berlin (1995)Google Scholar
  10. 10.
    Knapp, Anthony K.: Lie Group: Beyond An Introduction, 2nd edn. Birkhauser Boston, Basel, Berlin (2002)MATHGoogle Scholar
  11. 11.
    Perelomov, A.M.: Generalized Coherent States and Their Applications. Springer, Berlin (1986)MATHCrossRefGoogle Scholar
  12. 12.
    Rivasseau, V.: From Perturbative to Constructive Renormalization. Princeton University Press, Princeton (1991)Google Scholar
  13. 13.
    Rivasseau, V.: Quantum Spaces, Progress in Mathematical Physics, Birkhäuser Basel, 53, pp 19–107, (2007)Google Scholar
  14. 14.
    Rudin, W.: Quantum Spaces, Progress in Mathematical Physics, Birkhäuser Basel, 53, pp 19–107 (2007)Google Scholar
  15. 15.
    Szabo, R.J.: Quantum field theory on noncommutative spaces. Phys. Rept. 378, 207 (2003) [hep-th/0109162]MATHCrossRefGoogle Scholar
  16. 16.
    Wang, Z.: Construction of 2-dimensional Grosse–Wulkenhaar Model. arXiv:1104.3750 [math-ph]Google Scholar
  17. 17.
    Wang, Z.: Deformation quantization of rank I Bergman domain. arXiv:1110.2632 [math-ph]Google Scholar

Copyright information

© Springer Japan 2013

Authors and Affiliations

  1. 1.Laboratoire de Physique ThéoriqueCNRS UMR 8627, Université Paris XIOrsay CedexFrance

Personalised recommendations