Invariant Connections in Loop Quantum Gravity

Abstract

Given a group $G$ and an abelian $C^{\ast }$-algebra $\mathfrak{A}$, the antihomomorphisms $\Theta :G\to \mathrm{Aut}(\mathfrak{A})$ are in one-to-one with those left actions $\Phi :G\times \mathrm{Spec}(\mathfrak{A})\to \mathrm{Spec}(\mathfrak{A})$ whose translation maps ${\Phi }_g$ are continuous; whereby continuities of $\Theta$ and $\Phi$ turn out to be equivalent if $\mathfrak{A}$ is unital. In particular, a left action $\varphi :G\times X\to X$ can be uniquely extended to the spectrum of a $C^{\ast }$-subalgebra $\mathfrak{A}$ of the bounded functions on $X$ if ${\varphi }_g^{\ast }(\mathfrak{A})\subseteq \mathfrak{A}$ holds for each $g\in G$. In the present paper, we apply this to the framework of loop quantum gravity. We show that, on the level of the configuration spaces, quantization and reduction in general do not commute, i.e., that the symmetry-reduced quantum configuration space is (strictly) larger than the quantized configuration space of the reduced classical theory. Here, the quantum-reduced space has the advantage to be completely characterized by a simple algebraic relation, whereby the quantized reduced classical space is usually hard to compute.

33 pages. Revised version: Proof of Theorem 4.8 simplified; comments added to Sect. 1 and Sect. 5

References in corpus (6)

• On the Configuration Spaces of Homogeneous Loop Quantum Cosmology and Loop Quantum Gravity
• Embedding loop quantum cosmology without piecewise linearity
• Invariant Connections and Symmetry Reduction in Loop Quantum Gravity
• A Characterization of Invariant Connections
• Projective Structures in Loop Quantum Cosmology
• Uniqueness of Measures in Loop Quantum Cosmology

Cited by in corpus (12)

• Emergent Friedmann dynamics with a quantum bounce from quantum gravity condensates
• A quantum reduction to Bianchi I models in loop quantum gravity
• An elementary introduction to loop quantum gravity
• Minisuperspace models as infrared contributions
• An embedding of loop quantum cosmology in (b, v) variables into a full theory context
• An algebraic reduction of the `scaling gap' in the Navier-Stokes regularity problem
• Diffeomorphism invariant cosmological symmetry in full quantum gravity
• Kinematical uniqueness of homogeneous isotropic LQC
• Projective Structures in Loop Quantum Cosmology
• A Characterization of Invariant Connections
• Uniqueness of Measures in Loop Quantum Cosmology
• Symmetries of Analytic Curves