Regularization and Renormalization of Quantum Field Theories on Noncommutative Spaces

  • Harald Grosse
Conference paper
Part of the Lecture Notes in Physics book series (LNP, volume 616)


We first review regularization methods based on matrix geometry and show how an ultraviolet cut-off for scalar fields respecting symmetries results. Sections of bundles over the sphere can be quantized too.This procedure even allows to regularize supersymmetry without violating it.This work was extended recently to include quantum group covariant regularizations.

In a second part recent attempts to renormalize fourdimensional deformed quantum field theory models is reviewed. For scalar models the well-known IR-UV mixing does not allow to use standard techniques.The same applies to the Yang-Mills model in four dimensions.Only additional symmetry, as it occurs in the Wess-Zumino model, allows to avoid this problem.

Nevertheless there is some hope that the Yang-Mills model can be handled too. We used the Seiberg-Witten map to transform the noncommutative gauge field to a commutative one and used the degree of freedom of this map to obtain counter terms for the renormalization procedure. Finally a derivation of the Seiberg-Witten map from natural requirements is skechted.


Gauge Transformation Conformal Transformation Lorentz Transformation Noncommutative Geometry Gauge Parameter 
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.


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  1. 1.
    Grosse, H., and Madore, J., Phys. Lett., B283, 218 (1992)ADSMathSciNetGoogle Scholar
  2. 2.
    Doplicher, S., Fredenhagen, K., and Roberts, J.E., Commun. Math. Phys., 172, 187 (1995).zbMATHCrossRefADSMathSciNetGoogle Scholar
  3. 3.
    Connes, A., Noncommutative Geometry, Academic Press, San Diego 1994.zbMATHGoogle Scholar
  4. 4.
    J. Madore, An introduction to noncommutative differential geometry and its physical applications, Cambridge University Press, 1999.Google Scholar
  5. 5.
    J.M. Gracia-Bondia, J.C. Varilly, and H. Figueria, Elements of noncommutative geometry, Birkhäuser, Boston 2000.Google Scholar
  6. 6.
    H. Grosse, C. Klimcik, and P. Presnajder, Int.J.Theor.Phys. 35, 231 (1996), Commun.Math.Ph ys. 178, 507 (1996), Commun.Math.Ph ys. 185, 155 (1997).zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    H. Grosse, J. Madore, and H. Steinacker, J.Geom.Phys. 38, 208 (2001), hepth/ 0005273, hep-th/0103164.CrossRefMathSciNetGoogle Scholar
  8. 8.
    A.Yu. Alekseev, A. Recknagel, and V. Schomerus, JHEP 09, 023 (1999).CrossRefADSMathSciNetGoogle Scholar
  9. 9.
    N. Seiberg and E. Witten, JHEP 09, 032 (1999).CrossRefADSMathSciNetGoogle Scholar
  10. 10.
    A. Connes, M.R. Douglas, and A. Schwarz, JHEP 02, 003 (1998).CrossRefADSMathSciNetGoogle Scholar
  11. 11.
    T. Filk, Phys.Lett B 376, 53 (1996).CrossRefADSMathSciNetzbMATHGoogle Scholar
  12. 12.
    S. Minwalla, M. Van Raamsdonk, and N. Seiberg, JHEP 02, 002 (2000).Google Scholar
  13. 13.
    I. Chepelev and R. Roiban, JHEP 03, 001 (2001).CrossRefADSMathSciNetGoogle Scholar
  14. 14.
    A. Bichl et al, JHEP 10, 046 (2000).ADSGoogle Scholar
  15. 15.
    B. Jurčo, S. Schraml, P. Schupp and J. Wess, Eur. Phys.J. C 17, 521 (2000).CrossRefADSzbMATHGoogle Scholar
  16. 16.
    J. Madore, S. Schraml, P. Schupp and J. Wess, Eur. Phys. J. C 16, 161 (2000).CrossRefADSMathSciNetGoogle Scholar
  17. 17.
    B. Jurčo, P. Schupp and J. Wess, Nucl. Phys. B 604, 148 (2001).CrossRefADSzbMATHGoogle Scholar
  18. 18.
    R. Jackiw, Phys.Rev.Lett. 41, 1635 (1978).CrossRefADSGoogle Scholar
  19. 19.
    R. Jackiw, Acta Physica Austr.Suppl. XXII (1980) 383.Google Scholar
  20. 20.
    A. Gerhold, J. Grimstrup, H. Grosse, L. Popp, M. Schweda and R. Wulkenhaar, hep-th/0012112.Google Scholar
  21. 21.
    A. Bichl et al, JHEP 06, 013 (2001).CrossRefADSMathSciNetGoogle Scholar
  22. 22.
    A. Bichl et al, Eur.Phys. J. C 24, 165 (2002).zbMATHCrossRefADSGoogle Scholar

Copyright information

© Springer-VerlagBerlin Heidelber 2003

Authors and Affiliations

  • Harald Grosse
    • 1
  1. 1.University of ViennaAustria

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