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

TL;DR

This paper demonstrates that the Hilbert space basis used in the Ponzano-Regge-Turaev-Viro-Ooguri model for 3D quantum gravity aligns with the loop representation basis, and explores extensions to 4D quantum gravity.

Contribution

It establishes the equivalence of the combinatorial and loop representation bases in 3D quantum gravity and proposes a modified Regge calculus for 4D.

Findings

Non-degenerate geometries are contained in Witten's Hilbert space

Length computation in Witten's 3D gravity is demonstrated

A scalar product expression in 4D loop representation is provided

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.