Chern--Simons states in spin-network quantum gravity

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.

8 pages, RevTeX, 17 figures included

Cited by in corpus (15)

• Loop Quantum Gravity
• Quantum gravity with a positive cosmological constant
• A new vacuum for Loop Quantum Gravity
• Flux formulation of loop quantum gravity: Classical framework
• Chern-Simons theory, Stokes' Theorem, and the Duflo map
• Wave function of the Universe and Chern-Simons Perturbation Theory
• Vassiliev invariants: a new framework for quantum gravity
• The continuum approach to the BF vacuum: the U(1) case
• A finite spin-foam-based theory of three and four dimensional quantum gravity
• A new realization of quantum geometry
• A spin network generalization of the Jones Polynomial and Vassiliev invariants
• Taming Thiemann's Hamiltonian constraint in canonical loop quantum gravity: reversibility, eigenstates and graph-change analysis
• Abelian Chern-Simons theory, Stokes' theorem, and generalized connections
• Does loop quantum gravity imply Lambda=0?
• Computing the graph-changing dynamics of loop quantum gravity