The status of quantum geometry in the dynamical sector of loop quantum cosmology

TLDR: In this paper, the quantum geometry operators of LQC in both the kinematical and the physical Hilbert spaces of solutions to the quantum constraints are derived, and quantum geometry can be used to characterize the physical solutions, and the operators of quantum geometry preserve many of their kinematics.

Abstract: This paper is motivated by the recent papers by Dittrich and Thiemann and, respectively, Rovelli discussing the status of quantum geometry in the dynamical sector of quantum geometry. Since the papers consider model examples, we also study the issue in the case of an example, namely on the loop quantum cosmology model of a space-isotropic universe. We derive the Rovelli–Thiemann–Dittrich partial observables corresponding to the quantum geometry operators of LQC in both Hilbert spaces: the kinematical one and the physical Hilbert space of solutions to the quantum constraints. We find that quantum geometry can be used to characterize the physical solutions, and the operators of quantum geometry preserve many of their kinematical properties.