Semiclassical states in quantum gravity: Curvature associated to a Voronoi graph

TL;DR

This paper investigates how discrete Voronoi graph structures in loop quantum gravity can be used to recover classical smooth geometries, focusing on curvature computation and the compatibility of these graphs with continuous spaces.

Contribution

It introduces a mathematical framework using Voronoi graphs to analyze the emergence of classical geometry from quantum discrete structures in loop quantum gravity.

Findings

Voronoi graphs can be used to approximate classical curvature

Existing methods for curvature computation are tested and analyzed

Framework shows potential for connecting discrete quantum structures to smooth geometry

Abstract

The building blocks of a quantum theory of general relativity are expected to be discrete structures. Loop quantum gravity is formulated using a basis of spin networks (wave functions over oriented graphs with coloured edges), thus realizing the aforementioned expectation. Semiclassical states should, however, reproduce the classical smooth geometry in the appropriate limits. The question of how to recover a continuous geometry from these discrete structures is, therefore, relevant in this context. Following previous works by Bombelli et al. we explore this problem from a rather general mathematical perspective using, in particular, properties of Voronoi graphs to search for their compatible continuous geometries. We test the previously proposed methods for computing the curvature associated to such graphs and analyse the framework in detail, in the light of the results obtained.