Construction of the \Phi^4_4-quantum field theory on noncommutative Moyal space

We review our recent construction of the $\phi^4$-model on four-dimensional Moyal space. A milestone is the exact solution of the quartic matrix model $Z[E,J]=\int d\Phi \exp(tr(J\Phi- E\Phi^2 -(\lambda/4) \Phi^4))$ in terms of the solution of a non-linear equation for the 2-point function and the eigenvalues of $E$. The $\beta$-function vanishes identically. For the Moyal model, the theory of Carleman type singular integral equations reduces the construction to a fixed point problem. Its numerical solution reveals a second-order phase transition at $\lambda_c\approx-0.396$ and a phase transition of infinite order at $\lambda=0$. The resulting Schwinger functions in position space are symmetric and invariant under the full Euclidean group. They are only sensitive to diagonal matrix correlation functions, and clustering is violated. The Schwinger 2-point function is reflection positive iff the diagonal matrix 2-point function is a Stieltjes function. Numerically this seems to be the case for coupling constants $\lambda \in [\lambda_c,0]$.