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Renormalisation of \(\phi^4\)-theory on noncommutative \(\mathbb R^{4}\) in the matrix base. (English) Zbl 1075.82005
The papers shows that the real four-dimensional Euclidean noncommutative \(\phi^4\)-model is renormalisable to all orders in perturbation theory. The proof is hard, intricate and lengthy. A long introduction displayed its heuristics and its strategy. Loosely speaking one re-write the problem in the harmonic oscillator base of the Moyal plane and then one uses Meixner polynomials to solve the free theory. The direct calculation of the Feynman graphs is intractable, and to circumvent this hard task, one refers to a renormalization method based on flow equations which they have previously written for non-local (dynamical) matrix models. The matrix Polchinski equation is solved by using ribbon graphs drawn on Riemann surfaces.

82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
81T75 Noncommutative geometry methods in quantum field theory
81T08 Constructive quantum field theory
81T15 Perturbative methods of renormalization applied to problems in quantum field theory
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