  In quantum field theory (QFT) models, it often seems natural to use, instead
of wave functions from Fock space, wave functions that are not
square-integrable and have pre-factors involving divergent integrals (known as
infinite wave function renormalizations). We provide here a rigorous
implementation of this approach. Specifically, we define a certain space of
divergent integrals, a field extension $ eRen $ of the complex numbers
containing exponentials of such integrals, and certain vector spaces $
\overline{\mathscr{F}} \subset \overline{\mathscr{F}}_{ex} $ of wave functions
(not necessarily square-integrable) with pre-factors from $ eRen $. These
spaces extend a dense subspace of Fock space $ \mathscr{F} $. We apply this
construction to a class of non-relativistic QFT models with Yukawa-like
interaction, including the Van Hove model, Nelson's model, the Gross and the
Fr\"ohlich polaron. The formal (UV-divergent) Hamiltonian $ H_0 + A^\dagger + A
$ together with an infinite self-energy renormalization $ E_\infty $ and a mass
renormalization operator $ \delta m $ is rigorously implemented on $
\overline{\mathscr{F}} $. We construct a Weyl algebra over $ eRen $ that can be
represented by dressing transformations on a subspace of $
\overline{\mathscr{F}}_{ex} $. One such dressing transformation is used to pull
the Hamiltonian back to Fock space, giving it physical meaning.
Self-adjointness of the pulled-back Hamiltonian is proven for a dense Fock
space domain. This Hamiltonian involves pairwise interaction potentials and
does not suffer from a flat mass shell. For the polaron, the dynamics generated
by it differ from those obtained by cutoff renormalization.
