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

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.