Quantum gravity on polygons and $\Bbb R\times \Bbb Z_n$ FLRW model

TL;DR

This paper fully solves the quantum geometry of polygon graphs and explores an FLRW cosmological model on a discrete space, revealing quantum effects and particle creation phenomena.

Contribution

It provides a complete quantum geometric solution for $ ext{Z}_n$ polygons and applies it to a discrete FLRW model, analyzing quantum correlations and particle creation.

Findings

Quantum Levi-Civita connection is unique for $n e 4$.

Numerical correlation functions for quantum gravity on $ ext{Z}_n$.

Identified particle creation spectrum and no-particle creation conditions.

Abstract

We fully solve the quantum geometry of $\Bbb Z_n$ as a polygon graph with arbitrary metric lengths on the edges, finding a $*$-preserving quantum Levi-Civita connection which is unique for $n\ne 4$. As a first application, we numerically compute correlation functions for Euclideanised quantum gravity on $\Bbb Z_n$ for small $n$. We then study an FLRW model on $\Bbb R\times\Bbb Z_n$, finding the same expansion rate as for the classical flat FLRW model in 1+2 dimensions. We also look at particle creation on $\Bbb R\times \Bbb Z_n$ and find an additional $m=0$ adiabatic no particle creation expansion as well as the particle creation spectrum for a smoothed step expansion.