Reflection Positive Stochastic Processes Indexed by Lie Groups

TL;DR

This paper explores reflection positive Markov processes indexed by Lie groups, linking quantum field theory concepts with representation theory, and introduces new methods for constructing reflection positive unitary representations.

Contribution

It advances the understanding of reflection positivity in the context of Lie groups by developing new constructions of reflection positive unitary representations.

Findings

New constructions of reflection positive unitary representations

Analysis of measures on path spaces and Gaussian measures

Extension of reflection positivity to Lie group indexed processes

Abstract

Reflection positivity originates from one of the Osterwalder-Schrader axioms for constructive quantum field theory. It serves as a bridge between euclidean and relativistic quantum field theory. In mathematics, more specifically, in representation theory, it is related to the Cartan duality of symmetric Lie groups (Lie groups with an involution) and results in a transformation of a unitary representation of a symmetric Lie group to a unitary representation of its Cartan dual. In this article we continue our investigation of representation theoretic aspects of reflection positivity by discussing reflection positive Markov processes indexed by Lie groups, measures on path spaces, and invariant gaussian measures in spaces of distribution vectors. This provides new constructions of reflection positive unitary representations.