Spin Networks, Turaev-Viro Theory and the Loop Representation

TL;DR

This paper explores topological quantum field theories using spin networks and skein spaces, proposing new state space descriptions and inner products, and relating these to loop quantum gravity.

Contribution

It introduces a skein space-based description of Turaev-Viro and Ponzano-Regge state spaces, and connects these to the loop representation in quantum gravity.

Findings

Defined an inner product on skein space matching topological invariants

Provided a new skein space description of Turaev-Viro theory

Suggested a link between skein inner products and loop quantum gravity

Abstract

We investigate the Ponzano-Regge and Turaev-Viro topological field theories using spin networks and their $q$-deformed analogues. I propose a new description of the state space for the Turaev-Viro theory in terms of skein space, to which $q$-spin networks belong, and give a similar description of the Ponzano-Regge state space using spin networks. I give a definition of the inner product on the skein space and show that this corresponds to the topological inner product, defined as the manifold invariant for the union of two 3-manifolds. Finally, we look at the relation with the loop representation of quantum general relativity, due to Rovelli and Smolin, and suggest that the above inner product may define an inner product on the loop state space.