Renormalisation of Noncommutative ϕ 4-Theory by Multi-Scale Analysis
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.
KeywordsNeural Network Angular Momentum Span Tree Renormalisation Group Matrix Model
Unable to display preview. Download preview PDF.
- 5.Grosse, H., Wulkenhaar, R.: Renormalisation of ϕ 4-theory on noncommutative ℝ4 in the matrix base. Commun. Math. Phys. 256, no. 2, 305–374 (2005)Google Scholar
- 10.Rivasseau, V.: From perturbative to constructive renormalization. Princeton, NJ: Princeton Univ. Press, 1991Google Scholar
- 13.Grosse, H., Wulkenhaar, R.: Renormalisation of ϕ 4-theory on noncommutative ℝ4 to all orders. Lett. Math. Phys. 71, no. 1, 13–26 (2005)Google Scholar
- 14.Rivasseau, V., Vignes-Tourneret, F.: Non-commutative renormalization. http://arxiv.org/abs/ hep-th/0409312, 2004 to appear in Proceedings of conference ``Rigorous Quantum Field Theory'' in honor of J. Bros (19–21 July 2004, Saclay)Google Scholar
- 15.Connes, A., Douglas, M. R., Schwarz, A.: ``Noncommutative geometry and matrix theory: Compactification on tori,'' JHEP 9802, 003 (1998) [arXiv:hep-th/9711162]Google Scholar