Discretisation parameter and operator ordering in loop quantum cosmology with the cosmological constant

TL;DR

This paper investigates how discretisation parameters and operator ordering choices in loop quantum cosmology affect the resolution of the initial singularity and the recovery of classical behavior, emphasizing the importance of specific ordering rules.

Contribution

It identifies the operator orderings that eliminate the singularity and establishes a general ordering rule, clarifying key ambiguities in loop quantum cosmology with a cosmological constant.

Findings

Singularity avoidance depends on operator ordering.

A specific ordering rule is required for singularity removal.

Large volume limit recovers smooth wave function with fixed volume steps.

Abstract

In loop quantum cosmology, the Hamiltonian reduces to a finite difference operator. We study the initial singularity and the large volume limit against the ambiguities in the discretisation and the operator ordering within a homogeneous, isotropic and spatially flat model with the cosmological constant. We find that the absence of the singularity strongly depends on the choice of the operator ordering and the requirement for the absence singles out a very small class of orderings. Moreover we find a general ordering rule required for the absence of the singularity. We also find that the large volume limit naturally recovers a smooth wave function in the discretisation where each step corresponds to a fixed volume increment but not in the one where each step corresponds to a fixed area increment. If loop quantum cosmology is to be a phenomenological realisation of full loop quantum gravity, these results are important to fix the theoretical ambiguities.