Loop Quantization and Symmetry: Configuration Spaces

Abstract

Given two sets S₁, S₂ and unital C^*-algebras ${\mathfrak{A}_1}$, ${\mathfrak{A}_2}$ of functions thereon, we show that a map ${\sigma : {\bf S}_1 \longrightarrow {\bf S}_2}$ can be lifted to a continuous map ${\bar\sigma : {\rm spec}\, \mathfrak{A}_1 \longrightarrow {\rm spec}\, \mathfrak{A}_2}$ iff ${\sigma^\ast \mathfrak{A}_2 := \{\sigma^\ast f\, |\, f \in \mathfrak{A}_2\} \subseteq \mathfrak{A}_1}$. Moreover, ${\bar \sigma}$ is unique if existing, and injective iff ${\sigma^\ast \mathfrak{A}_2}$ is dense. Then, we apply these results to loop quantum gravity and loop quantum cosmology. For all usual technical conventions, we decide whether the cosmological quantum configuration space is embedded into the gravitational one; indeed, both are spectra of some C^*-algebras, say, ${\mathfrak{A}_{\rm cosm}}$ and ${\mathfrak{A}_{\rm grav}}$, respectively. Typically, there is no embedding, but one can always get an embedding by the defining ${\mathfrak{A}_{\rm cosm} := C^\ast(\sigma^\ast \mathfrak{A}_{\rm grav})}$, where ${\sigma}$ denotes the embedding between the classical configuration spaces. Finally, we explicitly determine ${C^\ast(\sigma^\ast \mathfrak{A}_{\rm grav})}$ in the homogeneous isotropic case for ${\mathfrak{A}_{\rm grav}}$ generated by the matrix functions of parallel transports along analytic paths. The cosmological quantum configuration space so equals the disjoint union of ${\mathbb{R}}$ and the Bohr compactification of ${\mathbb{R}}$, appropriately glued together.