Introducing quantum Ricci curvature

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. 1 arXiv:1712.08847v1 [hep-th]

Implementing quantum Ricci curvature

“…This seems to be the same short-distance effect we found for δ ≲ 5 for all of the piecewise flat geometries investigated in Ref. [9], irrespective of their behavior for larger δ, and which we have already identified as a discretization artifact. For δ ≳ 5, we enter a region of gentler, monotonic decrease, suggestive of a positive quantum Ricci curvature, corresponding to the initial section of the bottom curve in Fig.…”

“…II, we established in Ref. [9] that the same is not true in the realm of piecewise flat spaces, where we have identified c q ≡ ðd=δÞj δ¼5 . Using the same identification for the DT data, a relative shift between continuum and lattice data is needed to account for the different values of c q .…”

“…This implies that we are not interested in finite structures (like triangulations or other polygonizations of spacetime) per se, but only in suitable infinite limits associated with potential quantum gravity theories, where most lattice details will become irrelevant. An important outcome of our analysis of quantum Ricci curvature on various piecewise flat spaces [9] was that these so-called lattice artifacts appear to be confined to a region δ ≲ 5, with δ measured in discrete link units.…”

“…To improve on this situation, we recently introduced a new observable, the quantum Ricci curvature, and investigated some of its properties on smooth and piecewise flat classical spaces [9]. Building on these results, which will be summarized in Sec.…”

“…Our definition of quantum Ricci curvature [9] is inspired by Ollivier's coarse Ricci curvature [13], but uses a different notion of sphere distance, which is essential for the application in quantum gravity. 2 The crucial quantity to compute is the average sphere distanced of two overlapping spheres S δ p and S δ p 0 of radius δ, whose centers p and p 0 are a geodesic distance δ apart, as illustrated in Fig.…”