Chern--Simons states in spin-network quantum gravity

TL;DR

This paper explores the transformation of Chern--Simons states into the spin-network representation in quantum gravity, analyzing their properties and the action of the Hamiltonian constraint in a simplified setting.

Contribution

It introduces a method to transform Chern--Simons states into spin-network form and examines their behavior under the Hamiltonian constraint in a restricted class of networks.

Findings

The first two coefficients of the invariant expansion are annihilated by the Hamiltonian constraint.

The transformation is limited to trivalent nets with planar intersections.

Framing issues are identified and discussed.

Abstract

In the context of canonical quantum gravity in terms of Ashtekar's new variables, it is known that there exists a state that is annihilated by all the quantum constraints and that is given by the exponential of the Chern--Simons form constructed with the Asthekar connection. We make a first exploration of the transform of this state into the spin-network representation of quantum gravity. The discussion is limited to trivalent nets with planar intersections. We adapt an invariant of tangles to construct the transform and study the action of the Hamiltonian constraint on it. We show that the first two coefficients of the expansion of the invariant in terms of the inverse cosmological constant are annihilated by the Hamiltonian constraint. We also discuss issues of framing that arise in the construction.