Abstract
There have already been many attempts in the literature to modify the Wightman or Haag-Kastler axioms to be closer to the empirically successful Lagrangian approach to quantum field theory. It is suggested here that insofar as one fundamental difficulty of both Wightman and Lagrangian QFT is the postulate that the quantum field is an operator-valued distribution ---a linear map from a linear space of test functions to a linear space of operators---, we are motivated to consider taking a quantum field to be a *non*-linear map from a linear space of test functions to a linear space of operators, an approach that to my knowledge has not previously been proposed. Constructively, some of the non-linear possibilities for the scalar field case are introduced and discussed. The introduction of non-linearity widens the range of well-defined theories enough that they may provide worthwhile effective field models even if they cannot provide ultimately correct models. [I apologize that I cannot yet make the mathematical level be that of Nature or SciAm.]
Peter Warwick Morgan