Basis of the Ponzano-Regge-Turaev-Viro-Ooguri quantum-gravity model is the loop representation basis

In three dimensions (3D) the Ponzano-Regge-Turaev-Viro-Ooguri model provides a combinatorial definition of quantum gravity. The model is written in terms of a specific basis in the Hilbert spaces associated with the 2D boundaries of spacetime. We show that this basis is the same as the one that defines the loop representation of quantum gravity. We extend this construction to the physical 4D 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 4D. We discuss the general form that a nonperturbative quantum theory of gravity should have, and argue that this should be given by a generalization of Atiyah's topological quantum-field-theory axioms.