Combinatorial Quantum Gravity: Emergence of Geometric Space from Random Graphs

C Kelly,C A Trugenberger

doi:10.1088/1742-6596/1275/1/012016

Abstract

We review and extend the recently proposed model of combinatorial quantum gravity. Contrary to previous discrete approaches, this model is defined on (regular) random graphs and is driven by a purely combinatorial version of Ricci curvature, the Ollivier curvature, defined on generic metric spaces equipped with a Markov chain. It dispenses thus of notions such as simplicial complexes and Regge calculus and is ideally suited to extend quantum gravity to combinatorial structures which have a priori nothing to do with geometry. Indeed, our results show that geometry and general relativity emerge from random structures in a second-order phase transition due to the condensation of cycles on random graphs, a critical point that defines quantum gravity non-perturbatively according to asymptotic safety. In combinatorial quantum gravity the entropy area law emerges naturally as a consequence of infinite-dimensional critical behaviour on networks rather than on lattices. We propose thus that the entropy area law is a signature of the random graph nature of space-(time) on the smallest scales.

Highlights

Ultraviolet (UV) fixed points of statistical mechanics models define renormalizable quantum field theories via the Wilson renormalization group [1]
The asymptotic safety scenario [4] would be realized on networks, rather than traditional statistical mechanics models
In the traditional discrete approach to quantum gravity [5] a smooth background is assumed, which is approximated by piecewise flat geometries on which curvature is computed by Regge calculus [6]

Summary

INTRODUCTION

Ultraviolet (UV) fixed points of statistical mechanics models define renormalizable quantum field theories via the Wilson renormalization group [1]. When equipped with a Markov chain, like a probability measure, a purely combinatorial notion of Ricci curvature, first introduced by Ollivier [7–9], can be defined on such structures. This was used in [2] to define a model of purely combinatorial quantum gravity. This approach was subsequently pursued in [10], where a modified version of the Ollivier curvature was introduced. Gi where the sum runs over all the vertices of the graph and g is a coupling constant with dimension 1/action This expression for the Ollivier curvature still looks forbidding. As we will show in a moment, it will become extremely simple on the physical configuration space

CONFIGURATION SPACE AND COMBINATORIAL QUANTUM GRAVITY

THE CLASSICAL LIMIT AND THE MEAN FIELD ACTION