The LQC evolution operator of FRW universe with positive cosmological constant

TL;DR

This paper investigates the mathematical properties of the evolution operator in Loop Quantum Cosmology for a flat FRW universe with a positive cosmological constant, revealing conditions for self-adjointness and implications for physical states.

Contribution

It analyzes the self-adjointness of the evolution operator in LQC with positive cosmological constant and identifies a critical value affecting the spectrum and physical states.

Findings

For mbda<mbda_c, the operator has many self-adjoint extensions with discrete spectra.

For mbdambda_c, the operator is essentially self-adjoint but lacks physically interesting states.

The critical value mbda_c is approximately 10.3 times the Planck length squared.

Abstract

The self-adjointness of an evolution operator $\Theta_{\Lambda}$ corresponding to the model of flat FRW universe with massless scalar field and cosmological constant quantized in the framework of Loop Quantum Cosmology is studied in the case $\Lambda>0$. It is shown, that for $\Lambda<\Lambda_c\approx 10.3 \lPl^{-2}$ the operator admits many self-adjoint extensions, each of the purely discrete spectrum. On the other hand for $\Lambda\geq\Lambda_c$ the operator is essentially self-adjoint, however the physical Hilbert space of the model does not contain any physically interesting states.