$\theta$-splitting Densities and Reflection Positivity

TL;DR

This paper introduces a simple criterion for reflection positivity of measures related to Gaussian measures, extending lattice results to more abstract settings and allowing the construction of new reflection positive measures beyond lattice support.

Contribution

It provides a general condition for reflection positivity applicable to abstract measures, broadening the scope beyond traditional lattice-based measures.

Findings

Established a sufficient condition for reflection positivity.

Extended lattice reflection positivity results to abstract measure spaces.

Enabled construction of reflection positive measures outside lattice frameworks.

Abstract

A simple condition is given that is sufficient to determine whether a measure that is absolutely continuous with respect to a Gau{\ss}ian measure on the space of distributions is reflection positive. It readily generalises conventional lattice results to an abstract setting, enabling the construction of many reflection positive measures that are not supported on lattices.