Invariant Connections in Loop Quantum Gravity

Understand

Given a group $G$ and an abelian $C^*$-algebra $\mathfrak{A}$, the antihomomorphisms $\Theta\colon G\rightarrow \mathrm{Aut}(\mathfrak{A})$ are in one-to-one with those left actions $\Phi\colon G\times \mathrm{Spec}(\mathfrak{A})\rightarrow \mathrm{Spec}(\mathfrak{A})$ 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 $\phi\colon G \times X\rightarrow X$ can be uniquely extended to the spectrum of a $C^*$-subalgebra $\mathfrak{A}$ of the bounded functions on $X$ if $\phi_g^*(\mathfrak{A})\subseteq \mathfrak{A}$ 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.