On the space of generalized fluxes for loop quantum gravity

TL;DR

The paper addresses the problem of defining a momentum/flux space $\overline{\mathcal{E}}$ dual to the generalized connection space $\overline{\mathcal{A}}$ in loop quantum gravity. It analyzes a group Fourier-transform approach and finds that cylindrical consistency fails for the non-abelian gauge group $\mathrm{SU}(2)$, obstructing a straightforward projective-limit construction of fluxes. By abelianizing to $G=\mathrm{U}(1)$, the flux space emerges as an inductive limit and admits a pro-$C^*$-algebra description via pull-back, with explicit dual maps reflecting coarse-graining rules. The results imply that a new continuum-coarse-graining framework is needed for the full non-abelian flux variables and suggest a connection to the Bohr compactification approach used in loop quantum cosmology.

Abstract

We show that the space of generalized fluxes - momentum space - for loop quantum gravity cannot be constructed by Fourier transforming the projective limit construction of the space of generalized connections - position space - due to the non-abelianess of the gauge group SU(2). From the abelianization of SU(2), U(1)^3, we learn that the space of generalized fluxes turns out to be an inductive limit, and we determine the consistency conditions the fluxes should satisfy under coarse-graining of the underlying graphs. We comment on the applications to loop quantum cosmology, in particular, how the characterization of the Bohr compactification of the real line as a projective limit opens the way for a similar analysis for LQC.

Figures (2)

• Figure 1: The three elementary moves on graphs.
• Figure 2: Consistency conditions for fluxes across surfaces associated with the three elementary moves on graphs.

Theorems & Definitions (10)

• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Theorem A.1: rudin
• Theorem A.2
• proof
• Theorem A.3: Gel'fand representation, gelfand
• Theorem A.4
• proof