Moduli Structures, Separability of the Kinematic Hilbert Space and Frames in Loop Quantum Gravity

TL;DR

This paper introduces a novel approach using frames to address the separability of the kinematic Hilbert space in loop quantum gravity, reducing redundancy without extending the diffeomorphism group.

Contribution

It applies frame formalism to eliminate moduli space redundancy in high valence graphs, offering a new mathematical perspective in loop quantum gravity.

Findings

Reduces redundancy in the kinematic Hilbert space

Provides a new mathematical framework using frames

Avoids extending the diffeomorphism group

Abstract

We reassess the problem of separability of the kinematic Hilbert space in loop quantum gravity under a new mathematical point of view. We use the formalism of frames, a tool used in signal analysis, in order to remove the redundancy of the moduli structures in high valence graphs, without resorting to set extension of diffeomorphism group. For this, we introduce a local redundancy which encodes the concentration of frame vectors on the tangent spaces $T_pM$ around points of intersections $p$ of smooth loops $\alpha$ in $\mathbb{R}^{3}$.