In non-perturbative quantum field theory (QFT), formal calculations often yield wave functions that are not square-integrable and have pre-factors involving divergent integrals. We provide a rigorous construction of a vector space $ \overline{\mathscr{F}} $ which extends a dense subset of the Fock space $ \mathscr{F} $, but additionally contains those wave functions including divergent quantities. Moreover, we allow for a direct interpretation of divergent integrals as elements of a suitable vector space $ Ren_1 $ and we construct a field extension $ eRen $ of the complex numbers $ \mathbb{C} $ describing infinite wave function renormalizations. No cutoffs or regularizations are needed.

The extended state space construction is applied to a class of non-relativistic QFT models with Yukawa-like interaction, including the Van Hove model, Nelson's model, the Gross and the Fröhlich polaron. It is shown that these models amount exactly (not just approximately) to interactions by pair potentials, provided that a suitable dressing transformation is applied.

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