Consistent canonical quantization of general relativity in the space of Vassiliev knot invariants

TL;DR

This paper introduces a new approach to canonical quantum gravity by using Vassiliev knot invariants in the spin network representation, resulting in well-defined, finite constraints that replicate the classical algebra at the quantum level.

Contribution

It develops a consistent quantization framework for general relativity using Vassiliev knot invariants, ensuring finite and well-defined quantum constraints.

Findings

Constraints are finite and well-defined

Quantum constraints reproduce classical Poisson algebra

Method extends to 2+1 dimensions with correct quantum theory

Abstract

We present a quantization of the Hamiltonian and diffeomorphism constraint of canonical quantum gravity in the spin network representation. The novelty consists in considering a space of wavefunctions based on the Vassiliev knot invariants. The constraints are finite, well defined, and reproduce at the level of quantum commutators the Poisson algebra of constraints of the classical theory. A similar construction can be carried out in 2+1 dimensions leading to the correct quantum theory.