Consistent Canonical Quantization of General Relativity in the Space of Vassiliev Invariants

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 wave functions based on the Vassiliev 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.