Communications in Mathematical Physics

, Volume 256, Issue 2, pp 305–374 | Cite as

Renormalisation of ϕ4-Theory on Noncommutative ℝ4 in the Matrix Base

  • Harald Grosse
  • Raimar Wulkenhaar


We prove that the real four-dimensional Euclidean noncommutative ϕ4-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 ℝ4. 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.


Dispersion Relation Scalar Field Harmonic Oscillator Flow Equation Local Behaviour 
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© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.Institut für Theoretische PhysikUniversität WienWienAustria
  2. 2.Max-Planck-Institut für Mathematik in den NaturwissenschaftenLeipzigGermany

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