Conformal covariance and the split property

TL;DR

This paper proves that conformal covariance on the circle guarantees the split property for local nets of observables, but provides a counterexample in Minkowski space showing the split property does not always follow from diffeomorphism covariance.

Contribution

It establishes the automatic nature of the split property under full conformal covariance on the circle and presents a counterexample in Minkowski space.

Findings

Conformal covariance implies the split property on the circle.

Full diffeomorphism covariance is essential for this implication.

A counterexample exists in Minkowski space where covariance does not imply the split property.

Abstract

We show that for a conformal local net of observables on the circle, the split property is automatic. Both full conformal covariance (i.e. diffeomorphism covariance) and the circle-setting play essential roles in this fact, while by previously constructed examples it was already known that even on the circle, M\"obius covariance does not imply the split property. On the other hand, here we also provide an example of a local conformal net living on the two-dimensional Minkowski space, which - although being diffeomorphism covariant - does not have the split property.