Regular one-parameter groups, reflection positivity and their application to Hankel operators and standard subspaces

TL;DR

This paper extends classical results in algebraic quantum field theory by analyzing real regular one-parameter groups and their applications to Hankel operators and standard subspaces, removing the positivity constraint.

Contribution

It generalizes Borchers' and Longo-Witten's theorems to non-positive generators using the concept of real regular one-parameter groups.

Findings

Generalized classical theorems to broader frameworks

Established new connections with Hankel operators

Provided insights into standard subspaces without positivity constraints

Abstract

Standard subspaces are a well-studied object in algebraic quantum field theory (AQFT). Given a standard subspace ${\tt V}$ of a Hilbert space $\mathcal{H}$, one is interested in unitary one-parameter groups on $\mathcal{H}$ with $U_t {\tt V} \subseteq {\tt V}$ for every $t \in \mathbb{R}_+$. If $({\tt V},U)$ is a non-degenerate standard pair on $\mathcal{H}$, i.e. the self-adjoint infinitesimal generator of $U$ is a positive operator with trivial kernel, two classical results are given by Borchers' Theorem, relating non-degenerate standard pairs to positive energy representations of the affine group $\mathrm{Aff}(\mathbb{R})$ and the Longo-Witten Theorem, stating the the semigroup of unitary endomorphisms of ${\tt V}$ can be identified with the semigroup of symmetric operator-valued inner functions on the upper half-plane. In this thesis, we prove results similar to the theorems of Borchers and of Longo-Witten for a more general framework of unitary one-parameter groups without the assumption that their infinitesimal generator is positive. We replace this assumption by the weaker assumption that the triple $(\mathcal{H},{\tt V},U)$ is a so-called real regular one-parameter group.