# Reflection positivity and spectral theory

## TL;DR

This paper explores the spectral theory of reflection positivity in quantum physics, analyzing geometric and probabilistic properties, and establishing new theorems relating the Markov property to reflection positivity.

## Contribution

It provides a detailed comparison of spectral properties in reflection-positive Hilbert spaces and introduces two new theorems linking the Markov property to reflection positivity.

## Key findings

- OS-positivity can be expressed via projections and reflection operators
- The Markov property implies reflection positivity, but not vice versa
- Operators associated with OS-positive systems have canonical factorizations

## Abstract

We consider reflection-positivity (Osterwalder-Schrader positivity, O.S.-p.) as it is used in the study of renormalization questions in physics. In concrete cases, this refers to specific Hilbert spaces that arise before and after the reflection. Our focus is a comparative study of the associated spectral theory, now referring to the canonical operators in these two Hilbert spaces. Indeed, the inner product which produces the respective Hilbert spaces of quantum states changes, and comparisons are subtle. We analyze in detail a number of geometric and spectral theoretic properties connected with axiomatic reflection positivity, as well as their probabilistic counterparts; especially the role of the Markov property. This view also suggests two new theorems, which we prove. In rough outline: It is possible to express OS-positivity purely in terms of a triple of projections in a fixed Hilbert space, and a reflection operator. For such three projections, there is a related property, often referred to as the Markov property; and it is well known that the latter implies the former; i.e., when the reflection is given, then the Markov property implies O.S.-p., but not conversely. In this paper we shall prove two theorems which flesh out a much more precise relationship between the two. We show that for every OS-positive system $\left(E_{+},\theta\right)$, the operator $E_{+}\theta E_{+}$ has a canonical and universal factorization.