QFT on the non-commutative Moyal space and combinatorics

In this chapter we present the Phi? QFT model on the non-commutative Moyal space and the UV/IR mixing issue, which prevents it from being renormalizable. We then present the Grosse–Wulkenhaar Phi? QFT model on the non-commutative Moyal space, which changes the usual propagator of the Phi? model (based on the heat kernel formula) to a Mehler kernel based propagator. This Grosse–Wulkenhaar model is perturbatively renormalizable but it is not translation-invariant (translation-invariance being a usual property of high-energy physics models). We then show how the Mellin transform technique can be used to express the Feynman integrals of the Grosse-Wulkenhaar model. In the last part of the chapter, we present another Phi? QFT model on the non-commutative Moyal space, which is however both renormalizable and translation-invariant. We show the relation between the parametric representation of this model and the Bollobás–Riordan polynomial.