Extended State Space for describing renormalized Fock spaces in QFT

Abstract

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 F ⊂ F ex of wave functions (not necessarily square-integrable) with pre-factors from eRen. These spaces extend a dense subspace of Fock space F . We apply this construction to a class of non-relativistic QFT models with Yukawalike interaction, including the Van Hove model, Nelson's model, the Gross and the Fröhlich polaron. The formal (UV-divergent) Hamiltonian H 0 + A † + A together with an infinite self-energy renormalization E ∞ and a mass renormalization operator δm is rigorously implemented on F . We construct a Weyl algebra over eRen that can be represented by dressing transformations on a subspace of 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.