In the path integral approach to quantum gravity, the quantum amplitude for a transition between an initial spatial geometry and a final spatial geometry is obtained as a "sum over histories," where each "history" is a spacetime that interpolates between the chosen fixed initial and final spaces. If we knew how to perform this not-very-well-defined infinite sum, we could, in principle, compute anything we wanted to know, although the problems of interpretation might not disappear.
We don't know how to carry out this sum, though, and the usual approximations used in quantum field theory apparently fail -- that's what it means to say that general relativity is nonrenormalizable. An alternative is to approximate the path integral by putting the theory on a lattice, essentially breaking an infinite sum into a discrete, finite collection of "paths." For ordinary field theory on a lattice -- quantum chromodynamics, for example -- the lattice is held fixed, with fields placed at vertices or along edges. For general relativity, on the other hand, the lattice itself provides the dynamics. Just as flat triangles can be put together to form the curved surface of a geodesic dome, so flat simplices can be put together to form a curved spacetime.
The idea of this kind of discrete approach to quantum gravity is not at all new; it dates back at least to work by Martin Rocek and Ruth Williams in the early 1980s. Until recently, though, the resulting path integrals didn't seem to give anything close to a good approximation of a classical spacetime. Instead of a smooth, nearly flat spacetime, the results looked like either "crumpled" or "branched polymer" geometries, not at all like the world we live in. Recently, though, Jan Ambjorn, Renate Loll, and Jerzy Jurkiewicz have developed a new approach to the problem. Called "causal dynamical triangulations" (or "Lorentzian dynamical triangulations"), this program treats time in a new way, choosing new "gluing" rules that guarantee a well-behaved direction of time and, in the process, rule out certain kinds of quantum fluctuation of topology. While it is too soon to tell whether this approach will work, there are some very encouraging signs -- at least we seem to be able to obtain a good "nearly classical" four-dimensional spacetime from the simulations, along with what look to be the correct quantum fluctuations of spatial volume.
- Steve Carlip (Physics, UC Davis)