  There are still no interacting models of the Wightman axioms, suggesting that
the axioms are too tightly drawn. Here a weakening of linearity for quantum
fields is proposed, with the algebra still linear but with the quantum fields
no longer required to be tempered distributions, allowing explicit interacting
quantum field models. Interacting quantum fields should be understood to be
nonlinear quantum fields in this sense, because a set of effective field
theories encodes a dependence on the energy scale of measurement -- which is a
nontrivial property of the test functions -- so that correlation functions are
implicitly nonlinear functions of test functions in the conventional formalism.
In Local Quantum Physics terms, the algebraic models constructed here do not
satisfy the additivity property. Finite nonlinear deformations of quantized
electromagnetism are constructed as examples.
