Loop Quantization and Symmetry: Configuration Spaces

Details Loop Quantization and Symmetry: Configuration Spaces Loop Quantization and Symmetry: Configuration Spaces

2010-10-03

arxiv.org/abs/1010.0449v1

Physics

Mathematical Physics

33 pages, LaTeX

Scientific paper

Given two sets $S_1, S_2$ and unital C*-algebras $A_1$, $A_2$ of functions thereon, we show that a map $\sigma : S_1 --> S_2$ can be lifted to a continuous map $\bar\sigma : spec A_1 --> spec A_2$ iff $\sigma* A_2 := \{\sigma* f | f \in A_1\} \subset A_1$. Moreover, $\bar\sigma$ is unique if existing and injective iff $\sigma* A_2$ is dense. Next, we investigate the spectrum of the sum of two C*-algebras of functions on a locally compact $S_1$ having trivial intersection. In particular, for $A_0$ being the algebra of continuous functions vanishing at infinity and outside some open $Y \subset S_1$ with $A_0 A_1 \subset A_0$, the spectrum of $A_0 \dirvsum A_1$ equals the disjoint union of $Y$ and $spec A_1$, whereas the topologies of both sets are nontrivially interwoven. -- 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 where both are given as spectra of certain C*-algebras $A_\cosm$ and $A_\grav$. Typically, there is no embedding, but one can always get an embedding by the defining $A_\cosm := C*(\sigma* A_\grav)$, where $\sigma$ denotes the embedding between the classical configuration spaces. Finally, we explicitly determine $C*(\sigma* A_\grav)$ in the homogeneous isotropic case for $A_\grav$ generated by the matrix functions of parallel transports along analytic paths. The cosmological quantum configuration space so equals the disjoint union of $\R \setminus \{0\}$ and the Bohr compactification of $\R$, appropriately glued together.

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