  A nonlinear Wightman field is taken to be a nonlinear map from a linear space
of test functions to a linear space of Hilbert space operators, with
inessential modifications to other axioms only to the extent dictated by the
introduction of nonlinearity. Two approaches to nonlinear quantum fields are
constructed and discussed, the first of which, starting from Lagrangian QFT,
offers a fresh perspective on renormalization, while the second, starting from
linear Wightman fields, provides an extensive range of well-defined nonlinear
theories.
