Symmetry and Evolution in Quantum Gravity

TL;DR

This paper introduces a new operator constraint for quantum gravity that allows for genuine evolution, differing from Wheeler-DeWitt cosmology, and interprets local symmetries through a gauge theory framework based on spatial conformal diffeomorphisms.

Contribution

It proposes a novel quantum gravity framework with a well-defined notion of evolution and symmetry, based on conformal geometry and York time, distinct from traditional approaches.

Findings

The theory admits genuine evolution with an arbitrary parameter.

It provides a gauge theory interpretation of local symmetries.

The variational principle uniquely determines the constraints of the theory.

Abstract

We propose an operator constraint equation for the wavefunction of the Universe that admits genuine evolution. While the corresponding classical theory is equivalent to the canonical decomposition of General Relativity, the quantum theory makes predictions that are distinct from Wheeler-DeWitt cosmology. Furthermore, the local symmetry principle - and corresponding observables - of the theory have a direct interpretation in terms of a conventional gauge theory, where the gauge symmetry group is that of spatial conformal diffeomorphisms (that preserve the spatial volume of the Universe). The global evolution is in terms of an arbitrary parameter that serves only as an unobservable label for successive states of the Universe. Our proposal follows unambiguously from a suggestion of York whereby the independently specifiable initial data in the action principle of General Relativity is given by a conformal geometry and the spatial average of the York time on the spacelike hypersurfaces that bound the variation. Remarkably, such a variational principle uniquely selects the form of the constraints of the theory so that we can establish a precise notion of both symmetry and evolution in quantum gravity.