The basis of the Ponzano-Regge-Turaev-Viro-Ooguri model is the loop representation basis

TL;DR

The general form that a nonperturbative quantum theory of gravity should have is discussed, and it is argued that this should be given by a generalization of Atiyah's topological quantum-field-theory axioms.

Abstract

We show that the Hilbert space basis that defines the Ponzano-Regge- Turaev-Viro-Ooguri combinatorial definition of 3-d Quantum Gravity is the same as the one that defines the Loop Representation. We show how to compute lengths in Witten's 3-d gravity and how to reconstruct the 2-d geometry from a state of Witten's theory. We show that the non-degenerate geometries are contained in the Witten's Hilbert space. We sketch an extension of the combinatorial construction to the physical 4-d case, by defining a modification of Regge calculus in which areas, rather than lengths, are taken as independent variables. We provide an expression for the scalar product in the Loop representation in 4-d. We discuss the general form of a nonperturbative quantum theory of gravity, and argue that it should be given by a generalization of Atiyah's topological quantum field theories axioms.