Discrete approaches to quantum gravity in four dimensions

The construction of a consistent theory of quantum gravity is a problem in theoretical physics that has so far defied all attempts at resolution. One ansatz to try to obtain a non-trivial quantum theory proceeds via a discretization of space-time and the Einstein action. I review here three major areas of research: gauge-theoretic approaches, both in a path-integral and a Hamiltonian formulation, quantum Regge calculus, and the method of dynamical triangulations, confining attention to work that is strictly four-dimensional, strictly discrete, and strictly quantum in nature.Comment: 33 pages, invited contribution to Living Reviews in Relativity; the author welcomes any comments and suggestion

Four-dimensional dynamically triangulated gravity coupled to matter

We investigate the phase structure of four-dimensional quantum gravity coupled to Ising spins or Gaussian scalar fields by means of numerical simulations. The quantum gravity part is modelled by the summation over random simplicial manifolds, and the matter fields are located in the center of the 4-simplices, which constitute the building blocks of the manifolds. We find that the coupling between spin and geometry is weak away from the critical point of the Ising model. At the critical point there is clear coupling, which qualitatively agrees with that of gaussian fields coupled to gravity. In the case of pure gravity a transition between a phase with highly connected geometry and a phase with very ``dilute'' geometry has been observed earlier. The nature of this transition seems unaltered when matter fields are included. It was the hope that continuum physics could be extracted at the transition between the two types of geometries. The coupling to matter fields, at least in the form discussed in this paper, seems not to improve the scaling of the curvature at the transition point.Comment: 15 pages, 9 figures (available as PS-files by request). Late