Reflection positivity for the circle group

TL;DR

This paper characterizes certain unitary groups with Euclidean realizations via reflection positivity, linking modular theory, anti-unitary involutions, and KMS states in a mathematical framework.

Contribution

It provides a characterization of unitary one-parameter groups with Euclidean realizations through reflection positivity and anti-unitary involutions, connecting to modular theory and KMS states.

Findings

Characterization of unitary groups admitting Euclidean realizations

Connection between reflection positivity and anti-unitary involutions

Link between KMS states and reflection positivity on the circle

Abstract

In this note we characterize those unitary one-parameter groups U^c which admit euclidean realizations in the sense that they are obtained by the analytic continuation process corresponding to reflection positivity from a unitary representation $U$ of the circle group. These are precisely the ones for which there exists an anti-unitary involution $J$ commuting with $U^c$. This provides an interesting link with the modular data arising in Tomita--Takesaki theory. Introducing the concept of a positive definite function with values in the space of sesquilinear forms, we further establish a link between KMS states and reflection positivity on the circle.