Communications in Mathematical Physics

, Volume 262, Issue 3, pp 565–594 | Cite as

Renormalisation of Noncommutative ϕ 4-Theory by Multi-Scale Analysis

  • Vincent Rivasseau
  • Fabien Vignes-Tourneret
  • Raimar Wulkenhaar


In this paper we give a much more efficient proof that the real Euclidean ϕ 4-model on the four-dimensional Moyal plane is renormalisable to all orders. We prove rigorous bounds on the propagator which complete the previous renormalisation proof based on renormalisation group equations for non-local matrix models. On the other hand, our bounds permit a powerful multi-scale analysis of the resulting ribbon graphs. Here, the dual graphs play a particular rôle because the angular momentum conservation is conveniently represented in the dual picture. Choosing a spanning tree in the dual graph according to the scale attribution, we prove that the summation over the loop angular momenta can be performed at no cost so that the power-counting is reduced to the balance of the number of propagators versus the number of completely inner vertices in subgraphs of the dual graph.


Neural Network Angular Momentum Span Tree Renormalisation Group Matrix Model 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Vincent Rivasseau
    • 1
  • Fabien Vignes-Tourneret
    • 1
  • Raimar Wulkenhaar
    • 2
  1. 1.Laboratoire de Physique ThéoriqueUniversité Paris XIOrsay CedexFrance
  2. 2.Max-Planck-Institut für Mathematik in den NaturwissenschaftenGermany

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