Semiclassical limits of simplicial quantum gravity

Abstract

We consider the simplicial state sum model of Ponzano and Regge as a path integral for quantum gravity in three dimensions. We examine the Lorentzian geometry of a single 3-simplex and of a simplicial manifold, and interpret an asymptotic formula for 6j-symbols in terms of this geometry. This extends Ponzano and Regge's similar interpretation for Euclidian geometry. We give a geometric interpretation of the stationary points of this state sum, by showing that, at these points, the simplicial manifold may be mapped locally into flat Lorentzian or Euclidian space. This lends weight to the interpretation of the state sum as a path integral, which has solutions corresponding to both Lorentzian and Euclidian gravity in three dimensions.