Math Physics Seminar
We consider reflection-positivity (Osterwalder–Schrader positivity, OS-p.) as it is used in the study of renormalization questions in physics. This refers to specific Hilbert spaces that arise before and after reflection. Our focus is a comparative study of the associated spectral theory, now referring to the canonical operators in these two Hilbert spaces. Indeed, the inner product which produces the respective Hilbert spaces of quantum states changes, and comparisons are subtle. We analyze in detail a number of geometric and spectral theoretic properties connected with axiomatic reflection positivity, as well as their probabilistic counterparts; especially the role of the Markov property. This view also suggests two new theorems; in rough outline: It is possible to express OS-positivity purely in terms of a triple of projections in a fixed Hilbert space, and a reflection operator. For such three projections, there is a related property, often referred to as the Markov property; and it is well known that the latter implies the former; i.e., when the reflection is given, then the Markov property implies OS-p., but not conversely. In this paper we shall prove two theorems which flesh out a more precise relationship between the two.