Markov quantum fields on a manifold

TL;DR

This paper investigates scalar quantum field theory on compact manifolds, demonstrating the Markov property for free and interacting fields, and establishing reflection positivity and sewing operations, especially in two dimensions.

Contribution

It introduces the Markov property for quantum fields on manifolds and applies it to reflection positivity and sewing in two-dimensional cases, including interacting fields.

Findings

Markov property holds for free scalar fields with positive mass

Reflection positivity established for manifolds with reflection symmetry

Markov property extended to interacting fields in two dimensions

Abstract

We study scalar quantum field theory on a compact manifold. The free theory is defined in terms of functional integrals. For positive mass it is shown to have the Markov property in the sense of Nelson. This property is used to establish a reflection positivity result when the manifold has a reflection symmetry. In dimension d=2 we use the Markov property to establish a sewing operation for manifolds with boundary circles. Also in d=2 the Markov property is proved for interacting fields.