Approximating Connections in Loop Quantum Gravity

TL;DR

This paper develops a mathematical framework for approximating connections in loop quantum gravity, introducing a basis transformation and constructing semi-classical states that are peaked in both configuration and momentum variables.

Contribution

It derives the eigenbasis for the Wilson loop operator and shows how to construct states approximating classical connections, advancing semi-classical state construction in loop quantum gravity.

Findings

Eigenbasis for Wilson loop operator derived

Transformation between bases expressed via Chebyshev polynomials

Constructed states approximate classical connections on 3-manifolds

Abstract

We discuss the action of the configuration operators of loop quantum gravity. In particular, we derive the generalised eigenbasis for the Wilson loop operator and show that the transformation between this basis and the spin-network basis is given by an expansion in terms of Chebyshev polynomials. These results are used to construct states which approximate connections on the background 3-manifold in an analogous way that the weave states reproduce area and volumes of a given 3-metric. This should be necessary for the construction of genuine semi-classical states that are peaked both in the configuration and momentum variables.