  M{\o}ller maps are identifications between the observables of a
perturbatively interacting physical system and the observables of its
underlying free (i.e. non-interacting) system. This work studies and
characterizes obstructions to the existence of such identifications. The main
results are existence and importantly also non-existence theorems, which in
particular imply that M{\o}ller maps do not exist for non-Abelian Chern-Simons
and Yang-Mills theories on globally hyperbolic Lorentzian manifolds.
