Reflection positivity for the circle group

Abstract

In this note we characterize those unitary one-parameter groups U^c 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 $U^c$. 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.

16 pages; contribution to conference proceedings of "30th International Colloquium on Group Theoretical Methods in Physics; 14-18 July 2014, Gent, Belgium"