In this note we characterize those unitary one-parameter groups (Utc)t∈R which admit euclidean realizations in the sense that they are obtained by the analytic continuation process corresponding to reflection positivity from a unitary representation U of the circle group. These are precisely the ones for which there exists an anti-unitary involution J commuting with Uc. This provides an interesting link with the modular data arising in Tomita-Takesaki theory. Introducing the concept of a positive definite function with values in the space of sesquilinear forms, we further establish a link between KMS states and reflection positivity on the circle.
Publication Source (Journal or Book title)
Journal of Physics: Conference Series
Neeb, K., & Olafsson, G. (2015). Reflection positivity for the circle group. Journal of Physics: Conference Series, 597 (1) https://doi.org/10.1088/1742-6596/597/1/012004