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 JDelta 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.
تحميل البحث