2010/10/03 by Christian Fleischhack, Fleischhack, Christian · 1 citation
Mathematics · Physics and Astronomy · #34C27 #46L60 (Primary) 46L65 #53C05 #81T05 #83F05 #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #Mathematical Physics (math-ph) #gr-qc #math-ph #math.MP #msc:34C27 #msc:46L60 #msc:46L65 #msc:53C05 #msc:81T05 #msc:83F05
paper · pdf · doi:10.48550/arxiv.1010.0449
35 pages, LaTeX. Changes v1 to v2: algebra and spectrum for homogeneous isotropic case corrected (now Thm. 4.21; formerly 0 was missing in the spectrum); unitality assumption added in some parts of Sect. 2; other results basically not affected; presentation improved, including some reshuffling of subsections; former Sect. 3 extracted (enlarged version now as 1409.5273); Sect. 7, refs. added
arxiv created 2014/09/23 · arxiv updated 2014/09/25
Given two sets S1, S2 and unital C*-algebras A1, A2 of functions thereon, we show that a map σ: S1 \nach S2 can be lifted to a continuous map σ: \spec A1 → \spec A2 iff σ^∗ A2 := \σ^∗ f | f ∈ A2\ ⊂ A1. Moreover, σ is unique if existing, and injective iff σ^∗ A2 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^∗(σ^∗ A_\grav), where σ denotes the embedding between the classical configuration spaces. Finally, we explicitly determine C^∗(σ^∗ 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.