Lifting SU(2) Spin Networks to Projected Spin Networks

TL;DR

This paper investigates the relationship between projected spin networks used in quantum gravity models and standard SU(2) spin networks, revealing a non-unique mapping influenced by the Immirzi parameter and comparing their scalar products.

Contribution

It provides a detailed analysis of the map from projected spin networks to SU(2) spin networks, highlighting ambiguities and the role of the Immirzi parameter.

Findings

The map from projected to SU(2) spin networks is not one-to-one.

The ambiguity in the map is parameterized by the Immirzi parameter.

Scalar products differ between projected and SU(2) spin networks.

Abstract

Projected spin network states are the canonical basis of quantum states of geometry for the most recent EPR-FK spinfoam models for quantum gravity. They are functionals of both the Lorentz connection and the time normal field. We analyze in details the map from these projected spin networks to the standard SU(2) spin networks of loop quantum gravity. We show that this map is not one-to-one and that the corresponding ambiguity is parameterized by the Immirzi parameter. We conclude with a comparison of the scalar products between projected spin networks and SU(2) spin network states.