Renormalization of non-commutative Φ44 field theory in x space

In this paper we provide a new proof that the Grosse-Wulkenhaarn on-commutative scalar Phi(4)(4) theory is renormalizable to all orders in perturbation theory, and extend it to more general models with covariant derivatives. Our proof relies solely on a multiscale analysis in x space. We think this proof is simpler. It also allows direct interpretation in terms of the physical positions of the particles and should be more adapted to the future study of these theories (in particular at the non-perturbative or constructive level).