  We propose a partial answer to the question of what kind of ultrahigh-energy
physics has to be taken into account to circumvent the appearance of
ultraviolet divergencies; a more than sixty years old open question in quantum
electrodynamics. To this end we introduce a new theoretical concept to the
theory of quantum scattering: Perturbative S-matrices that depend on parameters
of a realistic regularization--realistic in the sense of Pauli and Villars
[Rev. Mod. Phys. 21, 434 (1949)]. We expect that these additional parameters
may provide some new information about the physics of quantum scattering. There
are such perturbative S-matrices also in the presence of non-renormalizable
interaction terms with no counterterms.
