Reflection positivity, duality, and spectral theory

We consider reflection-positivity (Osterwalder–Schrader positivity, OS-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 OS-p., but not conversely. In this paper we shall prove two theorems which flesh out a more precise relationship between the two. We show that for every OS positive system E+,θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( E_{+},\theta \right) $$\end{document}, the operator E+θE+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_{+}\theta E_{+}$$\end{document} has a canonical and universal factorization. Our second focus is a structure theory for all admissible reflections. Our theorems here are motivated by Phillips’ theory of dissipative extensions of unbounded operators. The word “Markov” traditionally makes reference to a random walk process where the Markov property in turn refers to past and future: expectation of the future, conditioned by the past. By contrast, our present initial definitions only make reference to three prescribed projection operators, and associated reflections. Initially, there is not even mention of an underlying probability space. This in fact only comes later.