Summary of discrete models for quantum gravity in three dimensions

• Main Author: Foxon, T. J
• Format: Dissertation
• Language: English
• Published: University of Cambridge 1994

We investigate discrete models for quantum gravity in three dimensions, based on topological quantum field theories. We begin by introducing the two main types of model which we shall investigate, namely Penrose's spin network model, and the Ponzano-Regge and Turaev-Viro simplicial state sum models. We briefly review the work of Ponzano and Regge showing how, in a certain semi-classical limit of their model, they cover three-dimensional Euclidean gravity. We go on to describe new work, by the author and J.W. Barrett, showing how the stationary points of a different semi-classical limit of the Ponzano-Regge partition function may be mapped to flat three-dimensional Lorentzian space, and consider how this partition function may be interpreted as a discrete version of a path integral for gravity in three dimensions. We describe the formalism of a topological quantum field theory. We examine the theory of Turaev and Viro, which is given by a simplicial state sum on a three-dimensional manifold, based on representations of the quantum group Uq(sl(2)). We show that it reduces to the Ponzano-Regge model in its q → 1 limit, and so it may be regarded as the naturally regularised version of that model. We describe new work investigating the relation between spin networks and simplicial state sum models. We show how the space of spin networks describes the state space for a two-dimensional surface in the Ponzano-Regge theory, and give a definition of the inner product on the state space which reproduces the topological inner product, defined by the union of two three-manifolds along their common boundary. We introduce Kauffman's q-deformed spin networks, and define the skein space, to which q-spin networks belong. We conjecture that a quotient of the skein space of a surface is isomorphic to the state space of the Turaev-Viro theory.