Lattice knot theory and quantum gravity in the loop representation

TL;DR

This paper develops a lattice-based loop representation of quantum gravity, demonstrating that key constraints reproduce classical algebraic structures and identifying knot invariants that solve the quantum constraints.

Contribution

It introduces a lattice implementation of the loop quantum gravity framework, establishing continuum limits and solving constraints using knot invariants from Chern--Simons theory.

Findings

Diffeomorphism constraint reproduces classical algebra in the continuum.

Certain knot invariants are solutions to the Hamiltonian constraint.

Regularization of the Hamiltonian constraint is achieved on the lattice.

Abstract

We present an implementation of the loop representation of quantum gravity on a square lattice. Instead of starting from a classical lattice theory, quantizing and introducing loops, we proceed backwards, setting up constraints in the lattice loop representation and showing that they have appropriate (singular) continuum limits and algebras. The diffeomorphism constraint reproduces the classical algebra in the continuum and has as solutions lattice analogues of usual knot invariants. We discuss some of the invariants stemming from Chern--Simons theory in the lattice context, including the issue of framing. We also present a regularization of the Hamiltonian constraint. We show that two knot invariants from Chern--Simons theory are annihilated by the Hamiltonian constraint through the use of their skein relations, including intersections. We also discuss the issue of intersections with kinks. This paper is the first step towards setting up the loop representation in a rigorous, computable setting.