Euclidean Quantum Field Theory on Commutative and Noncommutative Spaces

  • R. Wulkenhaar
Part of the Lecture Notes in Physics book series (LNP, volume 668)


I give an introduction to Euclidean quantum field theory from the point of view of statistical physics, with emphasis both on Feynman graphs and on the Wilson-Polchinski approach to renormalisation. In the second part I discuss attempts to renormalise quantum field theories on noncommutative spaces.


Partition Function Riemann Surface Noncommutative Geometry External Momentum 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.
    N. N. Bogolyubov, D. V. Shirkov, “Introduction to the theory of quantized fields,” Interscience (1959).Google Scholar
  2. 2.
    C. Itzykson, J.-B. Zuber, “Quantum field theory,” McGraw-Hill (1980).Google Scholar
  3. 3.
    K. G. Wilson, J. B. Kogut, “The Renormalization Group And The Epsilon Expansion,”Phys. Rept. 12, 75 (1974).CrossRefGoogle Scholar
  4. 4.
    J. Glimm, A. Jaffe, “Quantum Physics: a functional integral point of view,” Springer-Verlag (1981).Google Scholar
  5. 5.
    H. Grosse, “Models in statistical physics and quantum field theory,” Springer-Verlag (1988).Google Scholar
  6. 6.
    K. Osterwalder, R. Schrader, “Axioms For Euclidean Green’s Functions. I, II,”Commun. Math. Phys. 31, 83 (1973); 42, 281 (1975).CrossRefGoogle Scholar
  7. 7.
    G. Roepstorff, “Path integral approach to quantum physics: an introduction,” Springer-Verlag (1994).Google Scholar
  8. 8.
    B. Simon, “The P(Φ)2 Euclidean (Quantum) Field Theory,”Princeton University Press (1974).Google Scholar
  9. 9.
    G. Velo, A. S. Wightman (eds), “Renormalization Theory,”Reidel (1976).Google Scholar
  10. 10.
    W. Zimmermann, “Convergence Of Bogolyubov’s Method Of Renormalization In Momentum Space,”Commun. Math. Phys. 15, 208 (1969)[Lect. Notes Phys. 558, 217 (2000)].CrossRefGoogle Scholar
  11. 11.
    A. Connes, D. Kreimer, “Renormalization in quantum field theory and the Riemann-Hilbertproblem. I: The Hopf algebra structure of graphs and the main theorem,” Commun. Math Phys. 210, 249 (2000) [arXiv:hep-th/9912092].CrossRefGoogle Scholar
  12. 12.
    A. Connes, D. Kreimer, “Renormalization in quantum field theory and the Riemann-Hilbert problem. II: The beta-function, diffeomorphisms and the renormalization group,”Commun. Math. Phys. 216, 215 (2001) [arXiv:hep-th/0003188].Google Scholar
  13. 13.
    J. Polchinski,“Renormalization And Effective Lagrangians,”Nucl. Phys. B 231, 269 (1984).CrossRefGoogle Scholar
  14. 14.
    M. Salmhofer, “Renormalization: An Introduction,” Springer-Verlag (1998).Google Scholar
  15. 15.
    S. Doplicher, K. Fredenhagen, J. E. Roberts, “The Quantum structure of space-time at the Planck scale and quantum fields,” Commun. Math. Phys. 172, 187 (1995) [arXiv:hep-th/0303037].Google Scholar
  16. 16.
    E. Schrödinger, “XÜber die Unanwendbarkeit der Geometrie im Kleinen,” Naturwiss.31, 342 (1934).Google Scholar
  17. 17.
    A. Connes, “Noncommutative geometry,” Academic Press (1994).Google Scholar
  18. 18.
    S. Minwalla, M. Van Raamsdonk, N. Seiberg, “Noncommutative perturbative dynamics,” JHEP0002, 020 (2000) [arXiv:hep-th/9912072].CrossRefGoogle Scholar
  19. 19.
    N. Seiberg, E. Witten, “String theory and noncommutative geometry,” JHEP 9909, 032 (1999) [arXiv:hep-th/9908142].CrossRefGoogle Scholar
  20. 20.
    V. Gayral, J. M. Gracia-Bondía, B. Iochum, T. Schücker, J. C. Várilly, “Moyal planes are spectral triples,” Commun. Math. Phys. 246, 569 (2004) [arXiv:hep-th/0307241].CrossRefGoogle Scholar
  21. 21.
    I. Chepelev, R. Roiban, “Renormalization of quantum field theories on noncommutative Rd. I: Scalars,” JHEP 0005, 037 (2000) [arXiv:hep-th/9911098].CrossRefGoogle Scholar
  22. 22.
    T. Filk, “Divergencies In A Field Theory On Quantum Space,” Phys. Lett. B 376, 53 (1996).CrossRefGoogle Scholar
  23. 23.
    I. Chepelev, R. Roiban, “Convergence theorem for non-commutative Feynman graphs and renormalization,” JHEP 0103, 001 (2001) [arXiv:hep-th/0008090].CrossRefGoogle Scholar
  24. 24.
    Grosse, H., Wulkenhaar, R.: Renormalisation of 4 theory on noncommutative 4 to all orders. To appear in Lett. Math. Phys.,, 2004Google Scholar
  25. 25.
    H. Grosse, R. Wulkenhaar, “Renormalisation of Φ4 theory on noncommutative R2in the matrix base,” JHEP 0312, 019 (2003) [arXiv:hep-th/0307017].CrossRefGoogle Scholar
  26. 26.
    E. Langmann, R. J. Szabo, K. Zarembo, “Exact solution of noncommutative field theory in background magnetic fields,” Phys. Lett. B 569, 95 (2003) [arXiv:hep-th/0303082].CrossRefGoogle Scholar
  27. 27.
    E. Langmann, R. J. Szabo, K. Zarembo, “Exact solution of quantum field theory on noncommutative phase spaces,” JHEP 0401, 017 (2004) [arXiv:hep-th/0308043].CrossRefGoogle Scholar
  28. 28.
    Grosse, H.,Wulkenhaar, R.: The -function in duality-covariant noncommutative 4-theory. Eur. Phys. J. C 35, 277–282 (2004)Google Scholar
  29. 29.
    R. Koekoek, R. F. Swarttouw, “The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue,”arXiv:math.CA/9602214.Google Scholar
  30. 30.
    J. P. M. Luminet, J. Weeks, A. Riazuelo, R. Lehoucq, J. P. Uzan, “Dodecahedral space topology as an explanation for weak wide-angle temperature correlations in the cosmic microwave background,” Nature 425, 593 (2003) [arXiv:astro-ph/0310253].CrossRefPubMedGoogle Scholar

Authors and Affiliations

  • R. Wulkenhaar
    • 1
  1. 1.Max-Planck-Institute for Mathematics in the Sciences, Inselstraße 22-26LeipzigGermany

Personalised recommendations