Quantum gravity on polygons and $\mathbb{R}\times {\mathbb{Z}}_n$ FLRW model

Abstract

We fully solve the quantum geometry of ${\mathbb{Z}}_n$ as a polygon graph with arbitrary metric lengths on the edges, finding a $\ast$-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 ${\mathbb{Z}}_n$ for small $n$. We then study an FLRW model on $\mathbb{R}\times {\mathbb{Z}}_n$, finding the same expansion rate as for the classical flat FLRW model in 1+2 dimensions. We also look at particle creation on $\mathbb{R}\times {\mathbb{Z}}_n$ and find an additional $m=0$ adiabatic no particle creation expansion as well as the particle creation spectrum for a smoothed step expansion.

arXiv comment40 pages latex, 3 pdf figures. Version 1.6 added Sec 3.2 on the n-> infinity limit of Z_n as a classical circle. This Version 1.7 has only typo-level corrections relative to that

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• Quantum Riemannian geometry of the discrete interval and q-deformation
• Yang-Mills field from fuzzy sphere quantum Kaluza-Klein model