The general solution of the quantum Einstein equations?

TL;DR

This paper proposes a novel interpretation of the quantum Einstein equations in loop quantum gravity by relating the Hamiltonian constraint to skein relations on knots, ensuring solutions via knot invariants.

Contribution

It introduces a skein relation framework for the quantum Hamiltonian constraint, linking knot polynomials to solutions of quantum Einstein equations.

Findings

Knot polynomials compatible with skein relations solve quantum Einstein equations.

The approach maintains the consistency of the constraint algebra.

A specific invariant example demonstrates the method's viability.

Abstract

We suggest how to interpret the action of the quantum Hamiltonian constraint of general relativity in the loop representation as a skein relation on the space of knots. Therefore, by considering knot polynomials that are compatible with that skein relation, one guarantees that all the quantum Einstein equations are solved. We give a particular example of such invariant and discuss the consistency of the constraint algebra in this approach.