Canonical quantum gravity in the Vassiliev invariants arena: I. Kinematical structure

TL;DR

This paper extends Vassiliev invariants to spin networks, providing a new kinematical framework for canonical quantum gravity using knot invariants that are compatible with the constraints of the theory.

Contribution

It introduces a novel construction of Vassiliev invariants for spin networks, enabling the application of knot theory techniques to quantum gravity's kinematical structure.

Findings

Invariants are loop differentiable in the distributional sense.

Invariants are annihilated by the diffeomorphism constraint.

Provides a foundation for a consistent canonical quantum gravity theory.

Abstract

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.