Transition amplitudes and sewing properties for bosons on the Riemann sphere

TL;DR

This paper investigates scalar quantum fields on the sphere, demonstrating how correlation functions define trace class operators and establishing sewing properties in both massive and massless cases within a functional integral framework.

Contribution

It introduces a rigorous analysis of transition amplitudes and sewing properties for scalar fields on the sphere, including both massive and massless cases, using a functional integral approach.

Findings

Massive case: correlation functions define trace class operators.

Massless case: exponential fields form a conformal field theory.

Both cases: established sewing properties of the operators.

Abstract

We consider scalar quantum fields on the sphere, both massive and massless. In the massive case we show that the correlation functions define amplitudes which are trace class operators between tensor products of a fixed Hilbert space. We also establish certain sewing properties between these operators. In the massless case we consider exponential fields and have a conformal field theory. In this case the amplitudes are only bilinear forms but still we establish sewing properties. Our results are obtained in a functional integral framework.