  One of the axioms of quantum field theory is the property of unitarity of the
evolution operator. However, if one considers the quantum electrodynamics in
the external field in the leading order of perturbation theory, one will find
that the evolution transformation is a non-unitary canonical transformation of
creation and annihilation operators. This observation was one of the arguments
for the hypothesis that one should choose different representations of the
canonical commutation relations at different moments of time in the exact
quantum field theory. In this paper the contradiction is analyzed for the case
of a simple quantum mechanical model being an analog of the leading order of
the large-N field theory. On the one hand, this model is renormalized with the
help of the constructive field theory methods; the Hilbert space and unitary
evolution operator are constructed. On the other hand, the leading order of the
evolution transformation in the strong external field is shown to be
non-unitary. Thus, unitarity of evolution in the exact theory is not in
contradiction with non-unitarity of the approximate theory.
