KMS conditions, standard real subspaces and reflection positivity on the circle group

Karl-Hermann Neeb and Gestur Ólafsson

Vol. 299 (2019), No. 1, 117–169

DOI: 10.2140/pjm.2019.299.117

Abstract

We continue our investigations of the representation theoretic side of reflection positivity by studying positive definite functions $\psi$ on the additive group $\left(ℝ,+\right)$ satisfying a suitably defined KMS condition. These functions take values in the space $Bil\left(V\right)$ of bilinear forms on a real vector space $V$. As in quantum statistical mechanics, the KMS condition is defined in terms of an analytic continuation of $\psi$ to the strip

$$\left\{z\in ℂ0\le Imz\le \beta \right\}$$

with a coupling condition $\psi \left(i\beta +t\right)=\overline{\psi \left(t\right)}$ on the boundary. Our first main result consists of a characterization of these functions in terms of modular objects $\left(\Delta ,J\right)$ ($J$ an antilinear involution and $\Delta >0$ selfadjoint with $J\Delta J={\Delta }^{-1}$) and an integral representation.

Our second main result is the existence of a $Bil\left(V\right)$-valued positive definite function $f$ on the group ${ℝ}_{\tau }=ℝ⋊\left\{{id}_{ℝ},\tau \right\}$ with $\tau \left(t\right)=-t$ satisfying $f\left(t,\tau \right)=\psi \left(it\right)$ for $0\le t\le \beta$. We thus obtain a $2\beta$-periodic unitary one-parameter group on the GNS space ${\mathcal{\mathscr{H}}}_f$ for which the one-parameter group on the GNS space ${\mathcal{\mathscr{H}}}_{\psi }$ is obtained by Osterwalder–Schrader quantization.

Finally, we show that the building blocks of these representations arise from bundle-valued Sobolev spaces corresponding to the kernels

$${\left({\lambda }^2-d^2∕d{t}^2\right)}^{-1}$$

on the circle $ℝ∕\beta \mathbb{Z}$ of length $\beta$.

Keywords

KMS condition, reflection positivity, standard real subspace

Mathematical Subject Classification 2010

Primary: 43A35, 43A65, 47L30

Secondary: 47L90, 81T05

Milestones

Received: 31 October 2016

Accepted: 13 September 2018

Published: 18 April 2019