Analysing (cosmological) singularity avoidance in loop quantum gravity using U$(1)^3$ coherent states and Kummer's functions

TL;DR

This paper introduces a novel method using Kummer's functions to analyze singularity avoidance in loop quantum gravity, extending existing techniques to more complex graphs and fractional powers, and providing new insights into the inverse scale factor behavior.

Contribution

It develops a new procedure with Kummer's functions for handling fractional powers in LQG, extending formalisms to higher-valent graphs and improving estimates of quantum operators.

Findings

Correct classical limit for cubic graphs with linear q(r) powers

Higher order corrections can be systematically computed

Non-zero upper bound for inverse scale factor at singularity

Abstract

Using a new procedure based on Kummer's Confluent Hypergeometric Functions, we investigate the question of singularity avoidance in loop quantum gravity (LQG) in the context of U$(1)^3$ complexifier coherent states and compare obtained results with already existing ones. Our analysis focuses on the dynamical operators, denoted by q(r), whose products are the analogue of the inverse scale factor in LQG and also play a pivotal role for other dynamical operators such as matter Hamiltonians or the Hamiltonian constraint. For graphs of cubic topology and linear powers in q(r), we obtain the correct classical limit and demonstrate how higher order corrections can be computed with this method. This extends already existing techniques in the way how the involved fractional powers are handled. We also extend already existing formalisms to graphs with higher-valent vertices. For generic graphs and products of q(r), using estimates becomes inevitable and we investigate upper bounds for these semiclassical expectation values. Compared to existing results, our method allows to keep fractional powers involved in q(r) throughout the computations, which have been estimated by integer powers elsewhere. Similar to former results, we find a non-zero upper bound for the inverse scale factor at the initial singularity. Additionally, our findings provide some insights into properties and related implications of the results that arise when using estimates and can be used to look for improved estimates.