  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.
