Jones Polynomials for Intersecting Knots as Physical States of Quantum Gravity

TL;DR

This paper constructs explicit solutions to quantum gravity constraints using Jones polynomials for intersecting knots, linking knot theory with quantum gravity states in both connection and loop representations.

Contribution

It introduces a novel class of physical states in quantum gravity based on generalized Jones polynomials for intersecting knots, providing explicit forms in two representations.

Findings

Solutions to all quantum gravity constraints are expressed via generalized Jones polynomials.

The states correspond to nondegenerate spacetime metrics.

First explicit forms of quantum gravity states in both connection and loop representations.

Abstract

We find a consistent formulation of the constraints of Quantum Gravity with a cosmological constant in terms of the Ashtekar new variables in the connection representation, including the existence of a state that is a solution to all the constraints. This state is related to the Chern-Simons form constructed from the Ashtekar connection and has an associated metric in spacetime that is everywhere nondegenerate. We then transform this state to the loop representation and find solutions to all the constraint equations for intersecting loops. These states are given by suitable generalizations of the Jones knot polynomial for the case of intersecting knots. These are the first physical states of Quantum Gravity for which an explicit form is known both in the connection and loop representations. Implications of this result are also discussed.