Towards conformal loop quantum gravity

TL;DR

This paper develops a conformal extension of canonical gravity, integrating conformal and spin-gauge symmetries, offering a parameter-free formulation that advances loop quantum gravity and addresses key conceptual and technical issues.

Contribution

It introduces a conformal phase space extension of general relativity with new constraints and variables, providing a novel, parameter-free approach to loop quantum gravity.

Findings

Constructed a conformal geometrodynamics from extended ADM phase space.

Formulated a real spin connection variables framework with conformal symmetry.

Proposed a parameter-free formulation addressing the problem of time in quantum gravity.

Abstract

A discussion is given of recent developments in canonical gravity that assimilates the conformal analysis of gravitational degrees of freedom. The work is motivated by the problem of time in quantum gravity and is carried out at the metric and the triad levels. At the metric level, it is shown that by extending the Arnowitt-Deser-Misner (ADM) phase space of general relativity (GR), a conformal form of geometrodynamics can be constructed. In addition to the Hamiltonian and diffeomorphism constraints, an extra first class constraint is introduced to generate conformal transformations. This phase space consists of York's mean extrinsic curvature time, conformal three-metric and their momenta. At the triad level, the phase space of GR is further enlarged by incorporating spin-gauge as well as conformal symmetries. This leads to a canonical formulation of GR using a new set of real spin connection variables. The resulting gravitational constraints are first class, consisting of the Hamiltonian constraint and the canonical generators for spin-gauge and conformorphism transformations. The formulation has a remarkable feature of being parameter-free. Indeed, it is shown that a conformal parameter of the Barbero-Immirzi type can be absorbed by the conformal symmetry of the extended phase space. This gives rise to an alternative approach to loop quantum gravity that addresses both the conceptual problem of time and the technical problem of functional calculus in quantum gravity.