Semiclassical Quantum Gravity: Statistics of Combinatorial Riemannian Geometries

TL;DR

This paper develops a statistical framework to quantify uncertainties in the correspondence between discrete quantum spacetime structures and smooth classical geometries, aiding the understanding of semiclassical quantum gravity.

Contribution

It introduces a method using statistical geometry to estimate uncertainties in matching discrete structures to continuum geometries in semiclassical quantum gravity.

Findings

Proposes a statistical approach to quantify geometric uncertainties.

Demonstrates how to construct smooth geometries from discrete cell complexes.

Discusses combining geometric uncertainties with quantum state fluctuations.

Abstract

This paper is a contribution to the development of a framework, to be used in the context of semiclassical canonical quantum gravity, in which to frame questions about the correspondence between discrete spacetime structures at "quantum scales" and continuum, classical geometries at large scales. Such a correspondence can be meaningfully established when one has a "semiclassical" state in the underlying quantum gravity theory, and the uncertainties in the correspondence arise both from quantum fluctuations in this state and from the kinematical procedure of matching a smooth geometry to a discrete one. We focus on the latter type of uncertainty, and suggest the use of statistical geometry as a way to quantify it. With a cell complex as an example of discrete structure, we discuss how to construct quantities that define a smooth geometry, and how to estimate the associated uncertainties. We also comment briefly on how to combine our results with uncertainties in the underlying quantum state, and on their use when considering phenomenological aspects of quantum gravity.