Nonlinear dependence of the renormalization scale on test functions

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.