Physical time and other conceptual issues of QG on the example of LQC

TL;DR

This paper explores conceptual issues in quantum gravity using homogeneous isotropic Loop Quantum Cosmology, including quantum operators for proper time and spacetime metric, and analyzes singularity avoidance and differences in physical Hilbert spaces for different lapse choices.

Contribution

It introduces quantum operators for proper time and spacetime metric in LQC, compares solutions for different lapse functions, and studies singularity avoidance mechanisms.

Findings

Proper time is a quantum operator.

Quantum spacetime metric tensor operator is derived.

Singularity avoidance is supported by energy density operator analysis.

Abstract

Several conceptual aspects of quantum gravity are studied on the example of the homogeneous isotropic LQC model. In particular: $(i)$ The proper time of the co-moving observers is showed to be a quantum operator {and} a quantum spacetime metric tensor operator is derived. $(ii)$ Solutions of the quantum scalar constraint for two different choices of the lapse function are compared and contrasted. In particular it is shown that in case of model with masless scalar field and cosmological constant $\Lambda$ the physical Hilbert spaces constructed for two choices of lapse are the same for $\Lambda<0$ while they are significantly different for $\Lambda>0$. $(iii)$ The mechanism of the singularity avoidance is analyzed via detailed studies of an energy density operator, whose essential spectrum was shown to be an interval $[0,\rhoc]$, where $\rhoc\approx 0.41\rho_{\Pl}$. $(iv)$ The relation between the kinematical and the physical quantum geometry is discussed on the level of relation between observables.