Spin Networks, Turaev-Viro Theory and the Loop Representation

Details Spin Networks, Turaev-Viro Theory and the Loop Representation Spin Networks, Turaev-Viro Theory and the Loop Representation

1994-08-10

arxiv.org/abs/gr-qc/9408013v1

Class.Quant.Grav.12:951-964,1995

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

19 pages in Plain TeX, DAMTP-R-94/30

Scientific paper

10.1088/0264-9381/12/4/006

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.

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