Loop Quantization and Symmetry: Configuration Spaces

TL;DR

This paper establishes conditions for lifting maps between classical configuration spaces to their spectra in the context of C*-algebras, and applies these results to analyze embeddings of quantum configuration spaces in loop quantum gravity and cosmology.

Contribution

It provides a general criterion for lifting maps to spectra of C*-algebras and applies this to determine embeddings of cosmological into gravitational quantum configuration spaces.

Findings

A map between classical spaces lifts to spectra iff certain algebraic conditions hold.

Typically, cosmological and gravitational quantum spaces do not embed, but can be made to via algebraic modifications.

Explicit description of the cosmological quantum space as a union of nd the Bohr compactification.

Abstract

Given two sets $S_1, S_2$ and unital C*-algebras $A_1$, $A_2$ of functions thereon, we show that a map $\sigma : S_1 \nach S_2$ can be lifted to a continuous map $\bar\sigma : \spec A_1 \to \spec A_2$ iff $\sigma^\ast A_2 := \{\sigma^\ast f | f \in A_2\} \subset A_1$. Moreover, $\overline\sigma$ is unique if existing, and injective iff $\sigma^\ast A_2$ is dense. Then, we apply these results to loop quantum gravity and loop quantum cosmology. Here, the quantum configuration spaces are indeed spectra of certain C*-algebras $A_\cosm$ and $A_\grav$, respectively, whereas the choices for the algebras diverge in the literature. We decide now for all usual choices whether the respective cosmological quantum configuration space is embedded into the gravitational one. Typically, there is no embedding, but one can always get an embedding by defining $A_\cosm := C^\ast(\sigma^\ast A_\grav)$, where $\sigma$ denotes the embedding between the classical configuration spaces. Finally, we explicitly determine $C^\ast(\sigma^\ast A_\grav)$ in the homogeneous isotropic case for $A_\grav$ generated by the matrix functions of parallel transports along analytic paths. The cosmological quantum configuration space obtained this way, equals the disjoint union of $\R$ and the Bohr compactification of $\R$, appropriately glued together.