The non perturbative construction of quantum field models with nontrivial scattering in arbitrary dimension $d$ of the underlying Minkowski space-time is much more simple in the framework of quantum field theory with indefinite metric than in the positive metric case. In particular, there exist a number of solutions in the physical dimension $d=4$, where up to now no positive metric solutions are known. Here we review, why this is so, and we discuss some examples obtained by analytic continuation from the solutions of Euclidean covariant stochastic partial differential equations (SPDEs) driven by non-Gaussian white noise.