  For decades, a lot of work has been devoted to the problem of constructing a non-trivial quantum field theory in four-dimensional space time. This letter addresses the attempts to construct an algebraic quantum field theory in the framework of non-standard theories like hyperfunction or ultra-hyperfunction quantum field theory. For this purpose model theories of formally interacting neutral scalar fields are constructed and some of their characteristic properties like two-point functions are discussed. The formal self-couplings are obtained from local normally-ordered analytic redefinitions of the free scalar quantum field, mimicking a non-trivial structure of the resulting Lagrangians and equations of motion. 
Scalar models of formally interacting non-standard quantum fields in   Minkowski space-time