Conditional probabilities in Ponzano-Regge minisuperspace

TL;DR

This paper investigates how classical spacetime may emerge from quantum gravity in three dimensions by analyzing conditional probabilities and expectation values in the Ponzano-Regge model with the Hartle-Hawking initial state.

Contribution

It demonstrates that cutoff-independent expectation values arise under specific boundary conditions, linking quantum amplitudes to classical spacetime features.

Findings

Cutoff dependence of expectation values varies with boundary conditions.

Certain boundary constraints lead to cutoff-independent expectation values.

Results suggest a connection between boundary conditions and classical spacetime emergence.

Abstract

We examine the Hartle-Hawking no-boundary initial state for the Ponzano-Regge formulation of gravity in three dimensions. We consider the behavior of conditional probabilities and expectation values for geometrical quantities in this initial state for a simple minisuperspace model consisting of a two-parameter set of anisotropic geometries on a 2-sphere boundary. We find dependence on the cutoff used in the construction of Ponzano-Regge amplitudes for expectation values of edge lengths. However, these expectation values are cutoff independent when computed in certain, but not all, conditional probability distributions. Conditions that yield cutoff independent expectation values are those that constrain the boundary geometry to a finite range of edge lengths. We argue that such conditions have a correspondence to fixing a range of local time, as classically associated with the area of a surface for spatially closed cosmologies. Thus these results may hint at how classical spacetime emerges from quantum amplitudes.