Loop Quantum Cosmology II: Volume Operators

TL;DR

This paper constructs volume operators in loop quantum cosmology, explicitly solves for isotropic models, and explores how symmetry conditions influence the volume spectrum, providing insights into quantum cosmological implications.

Contribution

It introduces explicit volume operators for cosmological models in loop quantum gravity and analyzes their spectra under symmetry constraints, including a complete solution for isotropic cases.

Findings

Volume spectrum can be explicitly calculated for isotropic models.

Stronger symmetry conditions lead to smaller volume spectra.

Symmetry breaking causes level splitting in the spectrum.

Abstract

Volume operators measuring the total volume of space in a loop quantum theory of cosmological models are constructed. In the case of models with rotational symmetry an investigation of the Higgs constraint imposed on the reduced connection variables is necessary, a complete solution of which is given for isotropic models; in this case the volume spectrum can be calculated explicitly. It is observed that the stronger the symmetry conditions are the smaller is the volume spectrum, which can be interpreted as level splitting due to broken symmetries. Some implications for quantum cosmology are presented.