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

TL;DR

This paper investigates reflection positivity and KMS conditions for positive definite functions on the real line, characterizing them via modular objects, and explores their representation theory and connections to Sobolev spaces on the circle.

Contribution

It provides a characterization of KMS-satisfying functions through modular objects and establishes their representation as positive definite functions on a semidirect product group.

Findings

Characterization of KMS functions via modular objects and integral representation.

Existence of a positive definite function on a semidirect product group related to the KMS function.

Representation of these functions using bundle-valued Sobolev spaces on the circle.

Abstract

In the present paper we continue our investigations of the representation theoretic side of reflection positivity by studying positive definite functions \psi on the additive group (R,+) satisfying a suitably defined KMS condition. These functions take values in the space Bil(V) 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 { z \in C\: 0 \leq Im z \leq b} with a coupling condition \psi (ib + t) = \oline{\psi (t)} on the boundary. Our first main result consists of a characterization of these functions in terms of modular objects (\Delta, J) (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(V)-valued positive definite function f on the group R_\tau = R \rtimes {\id_\R,\tau} with \tau(t) = -t satisfying f(t,\tau) = \psi(it) for t \in R. We thus obtain a 2b-periodic unitary one-parameter group on the GNS space H_f for which the one-parameter group on the GNS space 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 1/(\lambda^2 - (d^2)/(dt^2}) on the circle R/bZ of length b.