Open FRW model in Loop Quantum Cosmology

TL;DR

This paper investigates the open Friedmann-Robertson-Walker model within Loop Quantum Cosmology, focusing on constructing the quantum scalar constraint operator and proving its self-adjointness for negative cosmological constants.

Contribution

It introduces a method to construct a suitable loop in hyperbolic geometry for the quantum scalar constraint in open LQC, extending previous formulations.

Findings

Constructed a loop in hyperbolic geometry for the $k=-1$ model.

Proved the quantum scalar constraint operator is essentially self-adjoint.

Extended the quantum scalar constraint to include negative cosmological constants.

Abstract

Open FRW model in Loop Quantum Cosmology is under consideration. The left and right invariant vector fields and holonomies along them are studied. It is shown that in the hyperbolic geometry of $k=-1$ it is possible to construct a suitable loop which provides us with quantum scalar constraint originally introduced by Vandersloot. The quantum scalar constraint operator with negative cosmological constant is proved to be essentially self-adjoint.