  The perturbative treatment of realistic quantum field theories, such as
quantum electrodynamics, requires the use of mathematical idealizations in the
approximation series for scattering amplitudes. Such mathematical idealisations
are necessary to derive empirically relevant models from the theory.
Mathematical idealizations can be either controlled or uncontrolled, depending
on whether current scientific knowledge can explain whether the effects of the
idealization are negligible or not. Drawing upon negative formal results in
asymptotic analysis (failure of Borel summability) and renormalization group
theory (failure of asymptotic safety), we argue that the mathematical
idealizations applied in perturbative quantum electrodynamics should be
understood as uncontrolled. This, in turn, leads to the problematic conclusion
that such theories do not have theoretical models in the standard understanding
of this term. The existence of unquestionable empirically successful theories
without theoretical models has significant implications both for our
understanding of the theory-model relationship in physics and the concept of
empirical adequacy.
