Canonical quantum gravity in the Vassiliev invariants arena. 1. Kinematical structure
Nov, 1999
21 pages
Published in:
- Class.Quant.Grav. 17 (2000) 3211-3238
e-Print:
- gr-qc/9911009 [gr-qc]
Report number:
- CGPG-99-11-1
View in:
Citations per year
Abstract: (arXiv)
We generalize the idea of Vassiliev invariants to the spin network context, with the aim of using these invariants as a kinematical arena for a canonical quantization of gravity. This paper presents a detailed construction of these invariants (both ambient and regular isotopic) requiring a significant elaboration based on the use of Chern-Simons perturbation theory which extends the work of Kauffman, Martin and Witten to four-valent networks. We show that this space of knot invariants has the crucial property -from the point of view of the quantization of gravity- of being loop differentiable in the sense of distributions. This allows the definition of diffeomorphism and Hamiltonian constraints. We show that the invariants are annihilated by the diffeomorphism constraint. In a companion paper we elaborate on the definition of a Hamiltonian constraint, discuss the constraint algebra, and show that the construction leads to a consistent theory of canonical quantum gravity.- quantum gravity: canonical
- loop space
- kinematics
- spin: network
- perturbation theory
- knot theory
- diffeomorphism: invariance
- Hamiltonian formalism
- algebra: constraint
- Chern-Simons term
References(34)
Figures(0)