Conformal Invariance and Renormalization Group in Quantum Gravity Near Two Dimensions

TL;DR

This paper explores quantum gravity near two dimensions, demonstrating how conformal invariance and renormalization group flow lead to a consistent theory that explains universe evolution and singularity resolution.

Contribution

It introduces a formulation of quantum gravity in $2+psilon$ dimensions that preserves volume-preserving diffeomorphisms and links covariance, conformal invariance, and RG flow to Einstein theory.

Findings

Long-distance universes require matter central charge 0<c<25

Big bang singularity is resolved by renormalization effects

Universes can bounce back from big crunch

Abstract

We study quantum gravity in $2+\epsilon$ dimensions in such a way to preserve the volume preserving diffeomorphism invariance. In such a formulation, we prove the following trinity: the general covariance, the conformal invariance and the renormalization group flow to Einstein theory at long distance. We emphasize that the consistent and macroscopic universes like our own can only exist for matter central charge $0<c<25$. We show that the spacetime singularity at the big bang is resolved by the renormalization effect and universes are found to bounce back from the big crunch. Our formulation may be viewed as a Ginzburg-Landau theory which can describe both the broken and the unbroken phase of quantum gravity and the phase transition between them.