Quantum Gravity and the Algebra of Tangles

TL;DR

This paper explores the algebraic structure of quantum gravity states represented by tangles, aiming to understand the *-algebra of observables and the inner product in loop quantum gravity using knot theory techniques.

Contribution

It introduces a tangle algebra acting on a larger solution space and constructs representations to determine the *-algebra structure and inner product in quantum gravity.

Findings

Constructed representations of the tangle algebra as quotients of the solution space.

Calculated the *-algebra structure using skein relations for the HOMFLY polynomial.

Determined the inner product for states related to the Jones polynomial.

Abstract

In Rovelli and Smolin's loop representation of nonperturbative quantum gravity in 4 dimensions, there is a space of solutions to the Hamiltonian constraint having as a basis isotopy classes of links in R^3. The physically correct inner product on this space of states is not yet known, or in other words, the *-algebra structure of the algebra of observables has not been determined. In order to approach this problem, we consider a larger space H of solutions of the Hamiltonian constraint, which has as a basis isotopy classes of tangles. A certain algebra T, the ``tangle algebra,'' acts as operators on H. The ``empty state'', corresponding to the class of the empty tangle, is conjectured to be a cyclic vector for T. We construct simpler representations of T as quotients of H by the skein relations for the HOMFLY polynomial, and calculate a *-algebra structure for T using these representations. We use this to determine the inner product of certain states of quantum gravity associated to the Jones polynomial (or more precisely, Kauffman bracket).