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

1993; American Physical Society; Volume: 48; Issue: 6 Linguagem: Inglês

10.1103/physrevd.48.2702

1538-4500

Carlo Rovelli,

Homotopy and Cohomology in Algebraic Topology

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.