Approximating Connections in Loop Quantum Gravity

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.

12 pages, 2 figures

References in corpus (3)

• Real Ashtekar Variables for Lorentzian Signature Space-times
• Weaving a classical geometry with quantum threads
• Loop Quantum Gravity on Non-Compact Spaces

Cited by in corpus (5)

• Gauge Field Theory Coherent States (GCS) : IV. Infinite Tensor Product and Thermodynamical Limit
• Statistical geometry of random weave states
• A proposal for analyzing the classical limit of kinematic loop gravity
• The graviton vacuum as a distributional state in kinematic Loop Quantum Gravity
• Bibliography of Publications related to Classical Self-dual variables and Loop Quantum Gravity