Vassiliev invariants: a new framework for quantum gravity

TL;DR

This paper introduces a novel approach to quantum gravity by utilizing Vassiliev invariants of knots, generalized to spin networks, which are loop differentiable and can serve as a foundation for defining quantum constraints.

Contribution

It demonstrates that Vassiliev invariants can be adapted to spin networks and used to rigorously define quantum gravity constraints as geometrical operators.

Findings

Vassiliev invariants are loop differentiable despite diffeomorphism invariance.

Explicit realization of the diffeomorphism constraint on Vassiliev invariants.

Proposals for constructing Hamiltonian constraints within this framework.

Abstract

We show that Vassiliev invariants of knots, appropriately generalized to the spin network context, are loop differentiable in spite of being diffeomorphism invariant. This opens the possibility of defining rigorously the constraints of quantum gravity as geometrical operators acting on the space of Vassiliev invariants of spin nets. We show how to explicitly realize the diffeomorphism constraint on this space and present proposals for the construction of Hamiltonian constraints.