A rigorous solution of the quantum Einstein equations

TL;DR

This paper demonstrates that the second coefficient of the Conway knot polynomial is annihilated by the Hamiltonian constraint in loop quantum gravity, using a regularized lattice approach that simplifies the calculation.

Contribution

It provides a rigorous lattice-based proof connecting knot invariants to quantum gravity constraints, advancing the mathematical understanding of quantum Einstein equations.

Findings

Second coefficient of Conway knot polynomial is annihilated by Hamiltonian constraint.

Lattice framework simplifies complex continuum calculations.

Explicit skein relations relate knot invariants to linking numbers.

Abstract

We show that the second coefficient of the Conway knot polynomial is annihilated by the Hamiltonian constraint of canonically quantized general relativity in the loop representation. The calculations are carried out in a fully regularized lattice framework. Crucial to the calculation is the explicit form of the skein relations of the second coefficient, which relate it to the Gauss linking number. Contrary to the lengthy formal continuum calculation, the rigorous lattice version can be summarized in a few pictures.