The Spectral Renormalization Flow Based on the Smooth Feshbach--Schur Map: The Introduction of the Semi-Group Property

TL;DR

This paper introduces a spectral renormalization flow compatible with the smooth Feshbach--Schur map, addressing an open problem and enhancing the mathematical tools used in spectral theory and quantum field theory.

Contribution

It develops a new spectral renormalization flow based on the smooth Feshbach--Schur map, improving compatibility and simplifying proofs in spectral analysis.

Findings

Introduces a spectral renormalization flow compatible with the smooth Feshbach--Schur map.

Addresses and solves an open problem from previous research.

Enhances the mathematical framework for spectral theory in quantum physics.

Abstract

The spectral renormalization method is a powerful mathematical tool that is prominently used in spectral theory in the context of low-energy quantum field theory and its original introduction in [5, 6] constituted a milestone in the field. Inspired by physics, this method is usually called renormalization group, even though it is not a group nor a semigroup (or, more properly, a flow). It was only in 2015 in [1] when a flow (or semigroup) structure was first introduced using an innovative definition of the renormalization of spectral parameters. The spectral renormalization flow in [1], however, is not compatible with the smooth Feshbach--Schur map (this is stated as an open problem in [1]), which is a lamentable weakness because its smoothness is a key feature that significantly simplifies the proofs and makes it the preferred tool in most of the literature. In this paper we solve this open problem introducing a spectral renormalization flow based on the smooth Feshbach--Schur map.