Area preserving transformationsin non-commutative space and NCCS theory

  • M. EliashviliEmail author
  • G. Tsitsishvili
theoretical physics


We propose a heuristic rule for the area transformation on the non-commutative plane. The non-commutative area preserving transformations are quantum deformations of the classical symplectic diffeomorphisms. The area preservation condition is formulated as a field equation in the non-commutative Chern-Simons gauge theory. A higher-dimensional generalization is suggested and the corresponding algebraic structure - the infinite-dimensional sin-Lie algebra - is extracted. As an illustrative example the second-quantized formulation for electrons in the lowest Landau level is considered.


Gauge Theory Field Equation Algebraic Structure Area Preserve Landau Level 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    R. Jackiw, Non-commuting fields and non-Abelian fluids, hep-th/0305027Google Scholar
  2. 2.
    R. Jackiw, S.-Y. Pi, Phys. Rev. Lett. 88, 111603 (2002)CrossRefGoogle Scholar
  3. 3.
    R. Jackiw, S.-Y. Pi, A. Polychronakos, Ann. Phys. (NY) 301, 157 (2002)CrossRefzbMATHGoogle Scholar
  4. 4.
    L. Susskind, The quantum Hall fluid and non-commutative Chern-Simons field theory, hep-th/0101029Google Scholar
  5. 5.
    M. Eliashvili, G. Tsitsishvili, Int. J. Mod. Phys. B 14, 1429 (2000)CrossRefMathSciNetGoogle Scholar
  6. 6.
    R.J. Szabo, Phys. Rep. 378, 207 (2003)CrossRefzbMATHGoogle Scholar
  7. 7.
    R. Jost, Rev. Mod. Phys. 36, 572 (1964)CrossRefGoogle Scholar
  8. 8.
    V.I. Arnold, Mathematical methods of classical mechanics (Springer-Verlag, Berlin 1978)Google Scholar
  9. 9.
    A.M. Perelomov, Generalized coherent states and their applications (Springer-Verlag, Berlin 1986)Google Scholar
  10. 10.
    J. Madore, An introduction to non-commutative geometry and its physical applications (Cambridge University Press 1999),Google Scholar
  11. 11.
    M. Eliashvili, G. Tsitsishvili, Int. J. Mod. Phys. B 16, 3725 (2002)CrossRefGoogle Scholar
  12. 12.
    S. Bahcall, L. Susskind, Int. J. Mod. Phys. B 5, 2735 (2002)Google Scholar
  13. 13.
    R. Manvelyan, R. Mkrtchyan, Phys. Lett. B 327, 47 (1994)CrossRefMathSciNetGoogle Scholar
  14. 14.
    D.B. Fairlie, P. Fletcher, C.K. Zachos, Phys. Lett. B 218, 203 (1989)CrossRefMathSciNetzbMATHGoogle Scholar
  15. 15.
    P. Dirac, Lectures on Quantum Mechanics (Belfer Graduate School of Science, Yeshiva University, New York 1964)Google Scholar
  16. 16.
    R. Jackiw, Nucl. Phys. Proc. Suppl. 108, 30 (2002)CrossRefzbMATHGoogle Scholar
  17. 17.
    N. Seiberg, E. Witten, JHEP 9909, 032 (1999)Google Scholar
  18. 18.
    S. Iso, D. Karabali, B. Sakita, Phys. Lett. B 196, 142 (1992)Google Scholar
  19. 19.
    J. Martinez, M. Stone, Int. J. Mod. Phys. B 7, 4389 (1993)MathSciNetGoogle Scholar
  20. 20.
    A. Cappelli, C. Trugenberger, G. Zemba, Nucl. Phys. B 396, 465 (1993)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Z.F. Ezawa, Quantum Hall effects (World Scientific, Singapore 2000)Google Scholar

Copyright information

© Springer-Verlag Berlin/Heidelberg 2003

Authors and Affiliations

  1. 1.Department of Theoretical PhysicsA. Razmadze Mathematical InstituteTbilisi Georgia

Personalised recommendations