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Renormalisation of ϕ^4-theory on noncommutative R^4 in the matrix base

Harald Grosse·Raimar Wulkenhaar
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Abstract

We prove that the real four-dimensional Euclidean noncommutative ϕ^4-model is renormalisable to all orders in perturbation theory. Compared with the commutative case, the bare action of relevant and marginal couplings contains necessarily an additional term: an harmonic oscillator potential for the free scalar field action. This entails a modified dispersion relation for the free theory, which becomes important at large distances (UV/IR-entanglement). The renormalisation proof relies on flow equations for the expansion coefficients of the effective action with respect to scalar fields written in the matrix base of the noncommutative R^4. The renormalisation flow depends on the topology of ribbon graphs and on the asymptotic and local behaviour of the propagator governed by orthogonal Meixner polynomials.

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Harald Grosse, Raimar Wulkenhaar. 2004-10-04. Renormalisation of ϕ^4-theory on noncommutative R^4 in the matrix base. https://doi.org/10.1007/s00220-004-1285-2

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