Is there any reasonable approach, essentially different from Wightman's axioms and Algebraic Quantum Field Theory, aimed at obtaining rigorous models for realistic Quantum Field Theories? (such as Quantum Electrodynamics).
EDIT: the reason for asking "essentially different" is the following. It is possible to intuitively think "states" as solutions of the equations of motion (in some sense, in a "multiparticle space"). In realistic interacting QFT, the equations of motion are nonlinear. So, according to my chosen intuitive concept, a reasonable state space should be nonlinear (something like a Hilbert manifold). Meanwhile, in Wightman or AQFT frameworks, state spaces are Hilbert spaces. This seems to correspond with the fact that it is very very difficult to construct interacting QFT's in these frameworks. So, as a desire... there should be a different, more interaction-friendly framework where realistic models arise in a more natural way.
Does something like in this direction already exist?