Renormalizable noncommutative quantum field theory

Abstract

We discuss the Euclidean noncommutative \({\phi^4_4}\)-quantum field theory as an example of a renormalizable field theory. Using a Ward identity, Disertori, Gurau, Magnen and Rivasseau were able to prove the vanishing of the beta function for the coupling constant to all orders in perturbation theory. We extend this work and obtain from the Schwinger–Dyson equation a non-linear integral equation for the renormalised two-point function alone. The non-trivial renormalised four-point function fulfils a linear integral equation with the inhomogeneity determined by the two-point function. These integral equations might be the starting point of a nonperturbative construction of a Euclidean quantum field theory on a noncommutative space. We expect to learn about renormalisation from this almost solvable model.