Renormalization of generalized spin-boson models with critical ultraviolet divergences

TL;DR

This paper rigorously constructs and analyzes generalized spin-boson models with critical UV divergences, establishing a consistent renormalization procedure and exploring cases requiring non-Fock representations.

Contribution

It introduces a direct renormalization method for models with critical UV divergences, including supercritical cases, and proves their equivalence to cut-off renormalized Hamiltonians.

Findings

Renormalized Hamiltonian matches cut-off renormalization results.

Supercritical cases require non-Fock representations and lead to trivial Hamiltonians.

Method applies to models with critical ultraviolet divergences.

Abstract

We provide a rigorous construction of generalized spin--boson models with commuting transition matrices and form factors exhibiting critical ultraviolet (UV) divergences. That is, we cover all divergences where a self-energy renormalization, but no non-Fock representation, is required. Our method is based on a direct definition of the renormalized Hamiltonian on a sufficiently large test domain, followed by a Friedrichs extension. We then prove that this Hamiltonian coincides with the one obtained by cut-off renormalization. Furthermore, we show that for specific supercritical cases, i.e., when a non-Fock representation is required, the renormalized Hamiltonian is trivial.