  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 prefactors involving divergent integrals (known as
infinite wave function renormalizations). Here, we rigorously construct vector
spaces containing divergent integrals, as well as two extended state vector
spaces, which contain a dense subspace of Fock space, but also incorporate
non-square-integrable wave functions with infinite wave function
renormalizations.
  As a demonstration, we apply this construction to a non-perturbative
renormalization of a class of simple non-relativistic QFT models, which are
polaron models with resting fermions. The Hamiltonian without cutoffs, an
infinite self-energy and a dressing transformation are defined as linear
operators on certain subspaces of the two Fock space extensions. This way, we
can obtain a renormalized Hamiltonian which can be realized as a densely
defined self-adjoint operator on Fock space.
