Probing the small distance structure of canonical quantum gravity using the conformal group

TL;DR

This paper explores a novel approach to canonical quantum gravity by integrating over the conformal factor first, aiming for a conformally invariant and finite effective theory, and discusses implications for unitarity and particle interactions.

Contribution

It proposes performing the conformal factor integral prior to other integrals, examines the effects of counter terms, and explores alternative ideas including the possibility of no counter terms, impacting quantum gravity understanding.

Findings

Effective theory may be conformally invariant and finite

Counter terms introduce unitarity issues and anomalies

A new perspective on particle-gravity interactions emerges

Abstract

In canonical quantum gravity, the formal functional integral includes an integration over the local conformal factor, and we propose to perform the functional integral over this factor before doing any of the other functional integrals. By construction, the resulting effective theory would be expected to be conformally invariant and therefore finite. However, also the conformal integral itself diverges, and the effects of a renormalization counter term are considered. It generates problems such as unitarity violation, due to a Landau-like ghost, and conformal anomalies. Adding (massive or massless) matter fields does not change the picture. Various alternative ideas are offered, including a more daring speculation, which is that no counter term should be allowed for at all. This has far-reaching and important consequences, which we discuss. A surprising picture emerges of quantized elementary particles interacting with a gravitational field, in particular gravitons, which are "partly classical". This approach was inspired by a search towards the reconciliation of Hawking radiation with unitarity and locality, and it offers basic new insights there.