Kinematical Foundations of Loop Quantum Cosmology

TL;DR

This paper reviews the mathematical foundations of loop quantum cosmology, compares different quantization approaches, and presents simplified proofs for key results on measures and symmetries in the theory.

Contribution

It provides a clear, simplified derivation of the non-commuting nature of quantization and symmetry reduction, and a uniqueness result for measures in loop quantum cosmology.

Findings

Quantization and symmetry reduction do not commute.

A uniqueness result for kinematical measures is established.

Simplified proofs for key results originally due to Hanusch.

Abstract

First, we review the $C^\ast$-algebraic foundations of loop quantization, in particular, the construction of quantum configuration spaces and the implementation of symmetries. Then, we apply these results to loop quantum gravity, focusing on the space of generalized connections and on measures thereon. Finally, we study the realm of homogeneous isotropic loop quantum cosmology: once viewed as the loop quantization of classical cosmology, once seen as the symmetric sector of loop quantum gravity. It will turn out that both theories differ, i.e., quantization and symmetry reduction do not commute. Moreover, we will present a uniqueness result for kinematical measures. These last two key results have originally been due to Hanusch; here, we give drastically simplified and direct proofs.