Non-commutative Renormalization

  • Vincent Rivasseau
Part of the Progress in Mathematical Physics book series (PMP, volume 53)


A new version of scale analysis and renormalization theory has been found on the non-commutative Moyal space. It could be useful for physics beyond the standard model or for standard physics in strong external fields. The good news is that quantum field theory is better behaved on non-commutative than on ordinary space: indeed it has no Landau ghost. We review this rapidly growing subject.


Ward Identity Power Counting Feynman Graph Multiscale Analysis Internal Line 
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]
    Neil J. Cornish, David N. Spergel, Glenn D. Starkman, and Eiichiro Komatsu, Constraining. the Topology of the Universe, Phys. Rev. Lett. 92 (2004), 201–302, astroph/0310233CrossRefMathSciNetGoogle Scholar
  2. [2]
    V. Rivasseau and F. Vignes-Tourneret, Non-Commutative Renormalization, in Rigorous Quantum Field Theory, a Festschrift for Jacques Bros, Birkhäuser Progress in Mathematics Vol. 251, 2007, hep-th/0409312.Google Scholar
  3. [3]
    M. Peskin and Daniel V. Schroeder (Contributor), An Introduction to Quantum Field. Theory, Perseus Publishing, (1995).Google Scholar
  4. [4]
    C. Itzykson and J.-B. Zuber, Quantum Field Theory, McGraw Hill, 1980.Google Scholar
  5. [5]
    P. Ramond, Field Theory, Addison-Wesley, 1994.Google Scholar
  6. [6]
    J. Glimm and A. Jaffe, Quantum Physics. A functional integral point of view. Mc-Graw and Hill, New York, 1981.zbMATHGoogle Scholar
  7. [7]
    Manfred Salmhofer, Renormalization: An Introduction, Texts and Monographs in Physics, Springer Verlag, 1999.Google Scholar
  8. [8]
    Renormalization, Poincar’e Seminar 2002, in Vacuum Energy Renormalization, Birkhäuser, Ed. by B. Duplantier and V. Rivasseau, Basel, 2003Google Scholar
  9. [9]
    V. Rivasseau, From Perturbative to Constructive Renormalization, Princeton University Press, 1991.Google Scholar
  10. [10]
    G. ’t Hooft and M. Veltman, Nucl. Phys. B50 (1972), 318.CrossRefGoogle Scholar
  11. [11]
    D. Gross and F. Wilczek, Ultraviolet Behavior of Non-Abelian Gauge Theories, Phys. Rev. Lett. 30 (1973), 1343–1346.CrossRefADSGoogle Scholar
  12. [12]
    H.D. Politzer, Reliable Perturbative Results for Strong Interactions?, Phys. Rev. Lett. 30 (1973), 1346–1349.CrossRefADSGoogle Scholar
  13. [13]
    D. Gross and F. Wilczek, Asymptotically Free Gauge Theories. I, Phys. Rev. D8 (1973), 3633–3652.ADSGoogle Scholar
  14. [14]
    K. Wilson, Renormalization group and critical phenomena, II Phase space cell analysis of critical behavior, Phys. Rev. B 4 (1974), 3184.CrossRefADSGoogle Scholar
  15. [15]
    J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, Oxford University Press, 2002.Google Scholar
  16. [16]
    G. Benfatto, G. Gallavotti, Renormalization Group, Princeton University Press, 1995.Google Scholar
  17. [17]
    G. Gentile and V. Mastropietro, Methods for the Analysis of the Lindstedt Series for KAM Tori and Renormalizability in Classical Mechanics. A Review with Some Applications, Rev. Math. Phys. 8 (1996).Google Scholar
  18. [18]
    W. Zimmermann, Convergence of Bogoliubov’s method for renormalization in momentum space, Comm. Math. Phys. 15 (1969), 208.zbMATHCrossRefADSMathSciNetGoogle Scholar
  19. [19]
    D. Kreimer, On the Hopf algebra structure of perturbative quantum field theories, Adv. Theor. Math. Phys. 2 (1998), 303–334, q-alg/9707029.zbMATHMathSciNetGoogle Scholar
  20. [20]
    A. Connes and D. Kreimer, Renormalization in quantum field theory and the Riemann-Hilbert problem i: The Hopf algebra structure of graphs and the main theorem, Commun. Math. Phys. 210 (2000), 249–273.zbMATHCrossRefADSMathSciNetGoogle Scholar
  21. [21]
    A. Connes and D. Kreimer, Renormalization in quantum field theory and the Riemann-Hilbert problem ii: The β-function, diffeomorphisms and the renormalization group, Commun. Math. Phys. 216 (2001), 215–241.zbMATHCrossRefADSMathSciNetGoogle Scholar
  22. [22]
    G. Gallavotti, Perturbation Theory, in Mathematical Physics towards the XXI century, p. 275–294, ed. R. Sen, A. Gersten, Ben Gurion University Press, Ber Sheva, 1994.Google Scholar
  23. [23]
    J. Feldman and E. Trubowitz, Perturbation Theory for Many Fermions Systems, Helv. Phys. Acta 63 (1990), 156.MathSciNetzbMATHGoogle Scholar
  24. [24]
    J. Feldman and E. Trubowitz, The Flow of an Electron-Phonon System to the Superconducting State, Helv. Phys. Acta 64 (1991), 213.MathSciNetGoogle Scholar
  25. [25]
    G. Benfatto and G. Gallavotti, Perturbation theory of the Fermi surface in a quantum liquid. A general quasi-particle formalism and one dimensional systems, Journ. Stat. Physics 59 (1990), 541.CrossRefMathSciNetzbMATHADSGoogle Scholar
  26. [26]
    J. Feldman, J. Magnen, V. Rivasseau and E. Trubowitz, An Intrinsic 1/N Expansion for Many Fermion Systems, Europhys. Letters 24 (1993), 437.zbMATHCrossRefADSMathSciNetGoogle Scholar
  27. [27]
    A.B. Zamolodchikov, JETP Letters 43 (1986), 731.ADSMathSciNetGoogle Scholar
  28. [28]
    K. Wiese, The functional renormalization group treatment of disordered systems, a review, Ann. Henri Poincaré 4 (2003), 473.MathSciNetCrossRefGoogle Scholar
  29. [29]
    Michael Green, John H. Schwarz and Edward Witten, Superstring theory, Cambridge University Press (1987).Google Scholar
  30. [30]
    A. Strominger and C. Vafa, Microscopic Origin of the Bekenstein-Hawking Entropy, Phys.Lett. B379 (1996), 99–104, hep-th/9601029.ADSMathSciNetGoogle Scholar
  31. [31]
    E. d’Hoker and T. Phong, Two-Loop Superstrings I, Main Formulas, Phys. Lett. B529 (2002), 241–255, hep-th/0110247.ADSMathSciNetGoogle Scholar
  32. [32]
    Lee Smolin, The Trouble With Physics: The Rise of String Theory, the Fall of a Science, and What Comes Next, Houghton-Mifflin, Sep. 2006.Google Scholar
  33. [33]
    H.S. Snyders, Quantized space-time, Phys. Rev 71 (1947), 38.CrossRefADSGoogle Scholar
  34. [34]
    A. Connes, M.R. Douglas, and A. Schwarz, Non-commutative geometry and matrix theory: Compactification on tori, JHEP 02 (1998), 003, hep-th/9711162.CrossRefADSMathSciNetGoogle Scholar
  35. [35]
    N. Seiberg and E. Witten, String theory and non-commutative geometry, JHEP 09 (1999), 032, hep-th/9908142.CrossRefADSMathSciNetGoogle Scholar
  36. [36]
    M.R. Douglas and N.A. Nekrasov, Non-commutative field theory, Rev. Mod. Phys. 73 (2001), 977–1029, hep-th/0106048.CrossRefADSMathSciNetGoogle Scholar
  37. [37]
    S. Minwalla, M. Van Raamsdonk, and N. Seiberg, Non-commutative perturbative dynamics, JHEP 02 (2000), 020, hep-th/9912072.CrossRefADSGoogle Scholar
  38. [38]
    H. Grosse and R. Wulkenhaar, Power-counting theorem for non-local matrix models and renormalization, Commun. Math. Phys. 254 (2005), no. 1, 91–127, hep-th/0305066.zbMATHCrossRefADSMathSciNetGoogle Scholar
  39. [39]
    H. Grosse and R. Wulkenhaar, Renormalization of ø 4-theory on non-commutative2 in the matrix base, JHEP 12 (2003), 019, hep-th/0307017.CrossRefADSMathSciNetGoogle Scholar
  40. [40]
    H. Grosse and R. Wulkenhaar, Renormalization of ø 4-theory on non-commutative4 in the matrix base, Commun. Math. Phys. 256 (2005), no. 2, 305–374, hep-th/0401128.zbMATHCrossRefADSMathSciNetGoogle Scholar
  41. [41]
    E. Langmann and R.J. Szabo, Duality in scalar field theory on non-commutative phase spaces, Phys. Lett. B533 (2002), 168–177, hep-th/0202039.ADSMathSciNetGoogle Scholar
  42. [42]
    V. Rivasseau, F. Vignes-Tourneret, and R. Wulkenhaar, Renormalization of non-commutative ø 4-theory by multi-scale analysis, Commun. Math. Phys. 262 (2006), 565–594, hep-th/0501036.zbMATHCrossRefADSMathSciNetGoogle Scholar
  43. [43]
    E. Langmann, R.J. Szabo, and K. Zarembo, Exact solution of quantum field theory on non-commutative phase spaces, JHEP 01 (2004), 017, hep-th/0308043.CrossRefADSMathSciNetGoogle Scholar
  44. [44]
    E. Langmann, R.J. Szabo, and K. Zarembo, Exact solution of non-commutative field theory in background magnetic fields, Phys. Lett. B569 (2003), 95–101, hep-th/0303082.ADSMathSciNetGoogle Scholar
  45. [45]
    E. Langmann, Interacting fermions on non-commutative spaces: Exactly solvable quantum field theories in 2n + 1 dimensions, Nucl. Phys. B654 (2003), 404–426, hep-th/0205287.CrossRefADSMathSciNetGoogle Scholar
  46. [46]
    J. Polchinski, Renormalization and Effective Lagrangians, Nucl. Phys. B231 (1984), 269.CrossRefADSGoogle Scholar
  47. [47]
    R. Gurau, J. Magnen, V. Rivasseau and F. Vignes-Tourneret, Renormalization of non-commutative ø 44 field theory in x space, Commun. Math. Phys. 267 (2006), no. 2, 515–542, hep-th/0512271.zbMATHCrossRefADSMathSciNetGoogle Scholar
  48. [48]
    I. Chepelev and R. Roiban, Convergence theorem for non-commutative Feynman graphs and renormalization, JHEP 03 (2001), 001, hep-th/0008090.CrossRefADSMathSciNetGoogle Scholar
  49. [49]
    I. Chepelev and R. Roiban, Renormalization of quantum field theories on noncommutatived. i: Scalars, JHEP 05 (2000), 037, hep-th/9911098.CrossRefADSMathSciNetGoogle Scholar
  50. [50]
    R. Gurau, V. Rivasseau, and F. Vignes-Tourneret, Propagators for non-commutative field theories, Ann. H. Poincaré 7 (2006), 1601–1628, hep-th/0512071.zbMATHCrossRefMathSciNetGoogle Scholar
  51. [51]
    F. Vignes-Tourneret, Renormalization of the orientable non-commutative Gross-Neveu model, Ann. H. Poincaré 8 (2007), math-ph/0606069.Google Scholar
  52. [52]
    H. Grosse and R. Wulkenhaar, The beta-function in duality-covariant non-commutative ø 4-theory, Eur. Phys. J. C35 (2004), 277–282, hep-th/0402093.ADSMathSciNetGoogle Scholar
  53. [53]
    K. Gawedzki and A. Kupiainen, Gross-Neveu model through convergent perturbation expansions, Comm. Math. Phys. 102 (1985), 1.CrossRefADSMathSciNetGoogle Scholar
  54. [54]
    J. Feldman, J. Magnen, V. Rivasseau and R. Sénéor, A renormalizable field theory: the massive Gross-Neveu model in two dimensions, Comm. Math. Phys. 103 (1986), 67.zbMATHCrossRefADSMathSciNetGoogle Scholar
  55. [55]
    K. Gawedzki and A. Kupiainen, Massless ø 44 theory: Rigorous control of a renormalizable asymptotically free model, Comm. Math. Phys. 99 (1985), 197.CrossRefADSMathSciNetGoogle Scholar
  56. [56]
    J. Feldman, J. Magnen, V. Rivasseau and R. Sénéor, Construction of infrared ø 44 by a phase space expansion, Comm. Math. Phys. 109 (1987), 437.CrossRefADSMathSciNetGoogle Scholar
  57. [57]
    M. Disertori and V. Rivasseau, Two and Three Loops Beta Function of Noncommutative Φ44 Theory, hep-th/0610224.Google Scholar
  58. [58]
    M. Disertori, R. Gurau, J. Magnen and V. Rivasseau, Vanishing of Beta Function of Non-commutative Φ44 to all orders, Submitted to Phys. Lett. B, hep-th/0612251.Google Scholar
  59. [59]
    W. Metzner and C. Di Castro, Conservation Laws and correlation functions in the Luttinger liquid, Phys. Rev. B 47 (1993), 16107.ADSGoogle Scholar
  60. [60]
    G. Benfatto and V. Mastropietro, Ward Identities and Chiral Anomaly in the Luttinger Liquid, Commun. Math. Phys. 258 (2005), 609–655.zbMATHCrossRefADSMathSciNetGoogle Scholar
  61. [61]
    G. Benfatto and V. Mastropietro, Ward Identities and Vanishing of the Beta Function for d = 1 Interacting Fermi Systems, Journal of Statistical Physics, 115 (2004), 143–184.CrossRefMathSciNetzbMATHADSGoogle Scholar
  62. [62]
    A. Sokal, An improvement of Watson’s theorem on Borel summability, Journ. Math. Phys. 21 (1980), 261.CrossRefADSMathSciNetGoogle Scholar
  63. [63]
    A. Abdesselam, A Complete Renormalization Group Trajectory Between Two Fixed Points, math-ph/0610018.Google Scholar
  64. [64]
    H. Grosse and H. Steinacker, Renormalization of the non-commutative ø 3 model through the Kontsevich model, Nucl. Phys. B746 (2006), 202–226, hep-th/0512203.CrossRefADSMathSciNetGoogle Scholar
  65. [65]
    H. Grosse and H. Steinacker, A nontrivial solvable non-commutative ø 3 model in 4 dimensions, JHEP 0608 (2006), 008, hep-th/0603052.CrossRefADSMathSciNetGoogle Scholar
  66. [66]
    H. Grosse, H. Steinacker, Exact renormalization of a non-commutative ø 3 model in 6 dimensions, hep-th/0607235.Google Scholar
  67. [67]
    Axel de Goursac, J.C. Wallet and R. Wulkenhaar, Non-commutative Induced Gauge Theory, hep-th/0703075.Google Scholar
  68. [68]
    R. Gurau and V. Rivasseau, Parametric representation of non-commutative field theory, to appear in Commun. Math. Phys, math-ph/0606030.Google Scholar
  69. [69]
    V. Rivasseau and A. Tanasa, Parametric representation of “critical” non-commutative QFT models, submitted to Commun. Math. Phys., hep-th/0701034.Google Scholar
  70. [70]
    R. Gurau and A. Tanasa, work in preparation.Google Scholar
  71. [71]
    L. Susskind, The Quantum Hall Fluid and Non-Commutative Chern Simons Theory, hep-th/0101029.Google Scholar
  72. [72]
    A. Polychronakos, Quantum Hall states as matrix Chern-Simons theory, JHEP 0104 (2001), 011, hep-th/0103013.CrossRefADSMathSciNetGoogle Scholar
  73. [73]
    S. Hellerman and M. Van Raamsdonk, Quantum Hall Physics = Non-commutative Field Theory, JHEP 0110 (2001), 039, hep-th/0103179.CrossRefADSGoogle Scholar
  74. [74]
    B. Duplantier, Conformal Random Geometry in Les Houches, Session LXXXIII, 2005, Mathematical Statistical Physics, A. Bovier, F. Dunlop, F. den Hollander, A. van Enter and J. Dalibard, eds., pp. 101–217, Elsevier B.V. (2006), math-ph/0608053.Google Scholar
  75. [75]
    G. ’t Hooft, A planar diagram theory for strong interactions, Nuclear Physics B 72 (1974), 461.ADSGoogle Scholar
  76. [76]
    V. Rivasseau, F. Vignes-Tourneret, Renormalization of non-commutative field theories, Luminy Lectures, hep-th/0702068Google Scholar
  77. [77]
    V. Rivasseau, An introduction to renormalization, in Poincaré Seminar 2002, ed. by B. Duplantier and V. Rivasseau. Progress in Mathematical Physics 30, Birkhäuser (2003), ISBN 3-7643-0579-7.Google Scholar
  78. [78]
    R. Feynman and A. Hibbs, Quantum Mechanics and Path Integrals, McGraw and Hill, New York 1965.zbMATHGoogle Scholar
  79. [79]
    C. Itzykson and J.M Drouffe, Statistical Field Theory, Volumes 1 and 2, Cambridge University Press 1991.Google Scholar
  80. [80]
    Giorgio Parisi, Statistical Field Theory, Perseus Publishing 1998.Google Scholar
  81. [81]
    J.P. Eckmann, J. Magnen and R. Sénéor, Decay properties and Borel summability for the Schwinger functions in P(ø)2 theories, Comm. Math. Phys. 39 (1975), 251.CrossRefADSGoogle Scholar
  82. [82]
    J. Magnen and R. Sénéor, Phase space cell expansion and Borel summability for the Euclidean ø 43 theory, Comm Math. Phys. 56 (1977), 237.CrossRefADSMathSciNetGoogle Scholar
  83. [83]
    C. Kopper Renormalization Theory based on Flow Equations, in Rigorous Quantum Field Theory, a Festschrift for Jacques Bros, Birkhäuser Progress in Mathematics Vol. 251, 2007, hep-th/0508143.Google Scholar
  84. [84]
    M. Bergère and Y.M.P. Lam, Bogoliubov-Parasiuk theorem in the α-parametric representation, Journ. Math. Phys. 17 (1976), 1546.CrossRefADSGoogle Scholar
  85. [85]
    C. de Calan and V. Rivasseau, Local existence of the Borel transform in Euclidean ø 44, Comm. Math. Phys. 82 (1981), 69.CrossRefADSMathSciNetGoogle Scholar
  86. [86]
    J.M. Gracia-Bondía and J. C. Várilly, Algebras of distributions suitable for phase space quantum mechanics. I, J. Math. Phys. 29 (1988), 869–879.zbMATHCrossRefADSMathSciNetGoogle Scholar
  87. [87]
    T. Filk, Divergencies in a field theory on quantum space, Phys. Lett. B376 (1996), 53–58.ADSMathSciNetGoogle Scholar
  88. [88]
    V. Gayral, Heat-kernel approach to UV/IR mixing on isospectral deformation manifolds, Annales Henri Poincaré 6 (2005), 991–1023, hep-th/0412233.zbMATHCrossRefMathSciNetADSGoogle Scholar
  89. [89]
    B. Simon, Functional integration and quantum physics, vol. 86 of Pure and applied mathematics. Academic Press, New York, 1979.Google Scholar
  90. [90]
    P.K. Mitter and P.H. Weisz, Asymptotic scale invariance in a massive Thirring model with U(n) symmetry, Phys. Rev. D8 (1973), 4410–4429.ADSGoogle Scholar
  91. [91]
    D.J. Gross and A. Neveu, Dynamical symmetry breaking in asymptotically free field theories, Phys. Rev. D10 (1974), 3235.ADSGoogle Scholar
  92. [92]
    C. Kopper, J. Magnen, and V. Rivasseau, Mass generation in the large N Gross-Neveu model, Commun. Math. Phys. 169 (1995), 121–180.zbMATHCrossRefADSMathSciNetGoogle Scholar
  93. [93]
    F. Vignes-Tourneret, Renormalization des théories de champs non-commutatives. Physique théorique, Université Paris 11, september, 2006, math-ph/0612014.Google Scholar
  94. [94]
    M. Disertori and V. Rivasseau, Continuous constructive fermionic renormalization, Annales Henri Poincaré 1 (2000), 1, hep-th/9802145.zbMATHCrossRefMathSciNetADSGoogle Scholar
  95. [95]
    A. Lakhoua, F. Vignes-Tourneret and J.C. Wallet, One-loop Beta Functions for the Orientable Non-commutative Gross-Neveu Model, submitted to JHEP, hep-th/0701170.Google Scholar
  96. [96]
    G. Kirchhoff, Über die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Verteilung galvanischer Ströme geführt wird, Ann. Physik Chemie 72(1847), 497–508.CrossRefADSGoogle Scholar
  97. [97]
    A. Abdesselam, Grassmann-Berezin Calculus and Theorems of the Matrix-Tree Type, math.CO/0306396.Google Scholar
  98. [98]
    C. de Calan and A.P.C. Malbouisson, Complete Mellin representation and asymptotic behaviours of Feynman amplitudes, Annales de l’Institut Henri Poincaré, physique théorique 32 (1980), 91–107.Google Scholar
  99. [99]
    C. de Calan, F. David and V. Rivasseau, Renormalization in the complete Mellin representation of Feynman amplitudes, Commun. Math. Phys. 78 (1981), 531–544.CrossRefADSGoogle Scholar
  100. [100]
    R. Gurau, A.P.C. Malbouisson, V. Rivasseau and A. Tanasa, Non-commutative Complete Mellin representation for Feynman Amplitudes, math-ph/0705.3437v1.Google Scholar

Copyright information

© Birkhäuser Verlag Basel 2007

Authors and Affiliations

  • Vincent Rivasseau
    • 1
  1. 1.Laboratoire de Physique Théorique, Bât. 210Université Paris XIOrsay CedexFrance

Personalised recommendations