Ultraviolet Renormalization of Spin Boson Models I. Normal and 2-Nilpotent Interactions
Benjamin Hinrichs, Jonas Lampart, Javier Valentín Martín
TL;DR
The paper resolves the UV renormalization problem for a broad class of generalized spin-boson models with normal or 2-nilpotent interactions. It couples a dressing transformation for normal interactions with an interior boundary condition approach for 2-nilpotent terms, establishing norm resolvent convergence of UV-regularized Hamiltonians to a well-defined renormalized limit H_{SB} and identifying the corresponding renormalization energies. A precise decomposition of the interaction into infrared, normal, and nilpotent parts, along with a robust IBC framework, yields rigorous selfadjoint realizations and convergence results under explicit conditions on the form factors. The work also delineates sharp non-renormalizability in the super-critical case, showing that certain UV divergences cannot be tamed by renormalization. Overall, the results provide a comprehensive, technique-bridging treatment of UV renormalization in spin-boson and related models, with potential implications for spectral stability and infrared behavior in the renormalized theories.
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.