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arXiv:2605.11725

A Feynman-Kac Formula for the Subcritical Ultraviolet-Renormalized Spin Boson Model

Daniel M. Fröhlich, Benjamin Hinrichs

Published May 12, 2026• 0 citations
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Abstract

We prove a Feynman-Kac formula (FKF) for the self-energy renormalized spin boson Hamiltonian, describing a two-state quantum system linearly coupled to a bosonic quantum field. Similar to recent FKFs for the Fröhlich polaron and the non- and semi-relativistic Nelson models, it yields a probabilistic treatment of the spin as a jump process, but treats the field on the usual bosonic Fock space. As an application, we prove that the existence of ground states for infrared-regular models persists the removal of an ultraviolet cutoff.

DOI
10.48550/arxiv.2605.11725

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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.
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We study the ultraviolet problem for models of a finite-dimensional quantum mechanical system linearly coupled to a bosonic quantum field, such as the (many-)spin boson model or its rotating-wave approximation. If the state change of the system upon emission or absorption of a boson is either given by a normal matrix or by a 2-nilpotent one, which is the case for the previously named examples, we prove an optimal renormalization result. We complement it, by proving the norm resolvent convergence of appropriately regularized models to the renormalized one. Our method consists of a dressing transformation argument in the normal case and an appropriate interior boundary condition for the 2-nilpotent case.