Vol. 299, No. 1, 2019

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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
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

$\left\{z\in ℂ\phantom{\rule{2.22144pt}{0ex}}0\le Imz\le \beta \right\}$

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 ${\mathsc{ℋ}}_{f}$ for which the one-parameter group on the GNS space ${\mathsc{ℋ}}_{\psi }$ 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 $ℝ∕\beta ℤ$ of length $\beta$.

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