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.
Highlights
THE CASE FOR QUANTUM OBSERVABLESA crucial ingredient for understanding the physics of nonperturbative quantum gravity are observables that capture the properties of spacetime in a diffeomorphisminvariant and background-independent manner, all the way down to the Planck scale
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
A crucial ingredient for understanding the physics of nonperturbative quantum gravity are observables that capture the properties of spacetime in a diffeomorphisminvariant and background-independent manner, all the way down to the Planck scale
Summary
A crucial ingredient for understanding the physics of nonperturbative quantum gravity are observables that capture the properties of spacetime in a diffeomorphisminvariant and background-independent manner, all the way down to the Planck scale. We will introduce a new geometric observable that has many of the desirable properties just described, a quasilocal quantity we call the “quantum Ricci curvature.” It is defined in purely geometric terms, without invoking any particular coordinate system, and has a welldefined classical meaning; we will construct it first on smooth Riemannian spaces. In nonperturbative quantum gravity models of the kind we are considering, integrated versions of the simplicial scalar curvature for D > 2 tend to be highly divergent in the continuum limit This happens because the density of the curvature defects grows as the lattice spacing a goes to zero, while the individual deficit angles do not “average out” on coarse-grained scales. Note that since it is natural to measure lengths and volumes in DT in terms of discrete units, measuring them is often reduced to counting, further simplifying implementation
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