Ultraviolet Renormalization of Spin Boson Models I. Normal and 2-Nilpotent Interactions

TL;DR

This paper investigates the ultraviolet renormalization of spin boson models with normal and 2-nilpotent interactions, establishing optimal renormalization results and norm resolvent convergence for these quantum field models.

Contribution

It provides the first rigorous renormalization results for spin boson models with these specific interaction types, using novel dressing transformations and boundary conditions.

Findings

Proves optimal ultraviolet renormalization for the models.

Establishes norm resolvent convergence of regularized models.

Introduces new methods for handling 2-nilpotent interactions.

Abstract

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.