Noncommutative quantum field theory

We summarize our recent construction of the φ4-model on four-dimensional Moyal space. This is achieved by solving the quartic matrix model for a general external matrix in terms of the solution of a non-linear equation for the 2-point function and the eigenvalues of that matrix. The β-function vanishes identically. For the Moyal model, the theory of Carleman type singular integral equations reduces the construction to a fixed point problem. The resulting Schwinger functions in position space are symmetric and invariant under the full Euclidean group. The Schwinger 2-point function is reflection positive iff the diagonal matrix 2-point function is a Stieltjes function. (Copyright copyright 2014 WILEY-VCH Verlag GmbH and Co. KGaA, Weinheim)