Communications in Mathematical Physics

, Volume 279, Issue 2, pp 355–379 | Cite as

Parametric Representation of “Covariant” Noncommutative QFT Models

  • Vincent Rivasseau
  • Adrian Tanasă


We extend the parametric representation of renormalizable non commutative quantum field theories to a class of theories which we call “covariant”, because their power counting is definitely more difficult to obtain.This class of theories is important since it includes gauge theories, which should be relevant for the quantum Hall effect.


Parametric Representation Vertex Versus Quantum Hall Effect Power Counting Feynman Graph 
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.
    Douglas M. and Nekrasov N. (2001). Noncommutative field theory. Rev. Mod. Phys. 73: 977–1029 CrossRefADSMathSciNetGoogle Scholar
  2. 2.
    Connes A., Douglas M.R. and Schwarz A. (1998). Noncommutative Geometry and Matrix Theory: Compactification on Tori. JHEP 9802: 3–43 CrossRefADSMathSciNetGoogle Scholar
  3. 3.
    Seiberg N. and Witten E. (1999). String theory and noncommutative geometry. JHEP 9909: 32–131 CrossRefADSMathSciNetGoogle Scholar
  4. 4.
    Susskind, L.: The Quantum Hall Fluid and Non-Commutative Chern Simons Theory.
  5. 5.
    Polychronakos A.P. (2001). Quantum Hall states on the cylinder as unitary matrix Chern-Simons theory. JHEP 06: 70–95 CrossRefADSMathSciNetGoogle Scholar
  6. 6.
    Hellerman S. and Van Raamsdonk M. (2001). Quantum Hall physics equals noncommutative field theory. JHEP 10: 39–51 CrossRefADSMathSciNetGoogle Scholar
  7. 7.
    Grosse H. and Wulkenhaar R. (2005). Power-counting theorem for non-local matrix models and renormalization. Commun. Math. Phys. 254: 91–127 zbMATHCrossRefADSMathSciNetGoogle Scholar
  8. 8.
    Grosse H. and Wulkenhaar R. (2005). Renormalizationof \(\phi^4\)-theory on noncommutative \({\mathbb R}^4\) in the matrix base. Commun. Math. Phys. 256: 305–374 zbMATHCrossRefADSMathSciNetGoogle Scholar
  9. 9.
    Rivasseau V., Vignes-Tourneret F. and Wulkenhaar R. (2006). Renormalization of noncommutative \(\phi^{\star 4}_4\)-theory by multi-scale analysis. Commun. Math. Phys. 262: 565–594 zbMATHCrossRefADSMathSciNetGoogle Scholar
  10. 10.
    Gurău R., Magnen J., Rivasseau V. and Vignes-Tourneret F. (2006). Renormalization of Non Commutative \(\Phi^4_4\) Field Theory in Direct Space. Commun. Math. Phys. 267: 515–542 CrossRefADSGoogle Scholar
  11. 11.
    Vignes-Tourneret F. (2007). Renormalization of the orientable non-commutative Gross-Neveu model. Ann. Henri Poincaré 8(3): 427–474 zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Langmann E., Szabo R.J. and Zarembo K. (2004). Exact solution of quantum field theory on noncommutative phase spaces. JHEP 0401: 17–87 CrossRefADSMathSciNetGoogle Scholar
  13. 13.
    Abdelmalek A. (2004). Grasmann-Berezin Calculus and Theorems of the Matrix-Tree Type. Adv. in Appl. Math. 33: 51–70 zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Gurău R. and Rivasseau V. (2007). Parametric representation of noncommutative field theory. Commun. Math. Phys. 272(3): 811–835 CrossRefADSGoogle Scholar
  15. 15.
    Itzkinson C. and Zuber J.-B. (1980). Quantum Field Theory. McGraw-Hill, New York Google Scholar
  16. 16.
    Rivasseau V. (1991). From perturbative to Constructive Field Theory. Princeton University Press, Princeton, NJ Google Scholar
  17. 17.
    Gurău R., Rivasseau V. and Vignes-Tourneret F. (2006). Propagators for Noncommutative Field Theories. Ann. Henri Poincaré 7(7–8): 1601–1628 CrossRefGoogle Scholar
  18. 18.
    Filk T. (1996). Divergencies in a field theory on quantum space. Phys. Lett. B 376: 53–58 CrossRefADSMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Laboratoire de Physique Théorique, bât. 210CNRS UMR 8627, Université Paris XIOrsay CedexFrance

Personalised recommendations