  Quantum electrodynamics exhibits an informal nonlinear dependence on Lorentz
invariant test function properties that determine the renormalization scale,
such as Mandelstam variables, contrary to the linear dependence on test
functions that is required by the Wightman axioms. A first example of an
alternative interacting quantum field formalism that has a comparable weakly
nonlinear dependence on Dirac spinor test functions is constructed, using
U(1)-gauge connections and U(1)-gauge invariant Dirac spinor test functions.
