Introducing Quantum Ricci Curvature

TL;DR

This paper introduces a new notion of quantum Ricci curvature designed for non-smooth and discrete spaces, enabling geometric analysis in quantum gravity contexts with scalable and computable properties.

Contribution

It defines a coarse-grained Ricci curvature applicable to non-smooth spaces, tested on various models, bridging classical and quantum geometric analysis.

Findings

Reproduces classical curvature characteristics on large scales

Shows good averaging properties on triangulated spaces

Behaves consistently across different discrete models

Abstract

Motivated by the search for geometric observables in nonperturbative quantum gravity, we define a notion of coarse-grained Ricci curvature. It is based on a particular way of extracting the local Ricci curvature of a smooth Riemannian manifold by comparing the distance between pairs of spheres with that of their centres. The quantum Ricci curvature is designed for use on non-smooth and discrete metric spaces, and to satisfy the key criteria of scalability and computability. We test the prescription on a variety of regular and random piecewise flat spaces, mostly in two dimensions. This enables us to quantify its behaviour for short lattices distances and compare its large-scale behaviour with that of constantly curved model spaces. On the triangulated spaces considered, the quantum Ricci curvature has good averaging properties and reproduces classical characteristics on scales large compared to the discretization scale.