Quantum mechanics of the gravitational field

An approach to the quantum theory of gravitation is developed by analogy with the quantum mechanics of the simplest generally covariant system—the relativistic point particle. The central object in the formalism is the transition amplitude from one three-geometry to another which is given by a path integral. In that path integral one sums over all possible histories which connect two three-geometries separated by a given local proper time and then integrates over all possible proper-time separations. The choice of the range of integration for the proper time fixes the boundary conditions for the transition amplitude. If only positive proper times are allowed, the resulting amplitude is causal. A perturbation theory is developed in which the expansion parameter is the signature which takes the value minus one when the field histories (spacetimes) have hyperbolic signature and plus one for the Eclidean case. The "free" theory corresponds to zero signature and may be viewed as the result of replacing the Lorentz group as a symmetry group of the tangent spaces by one of its contractions, namely that one where the speed of light approaches zero. It is argued that besides the processes in which the universe starts or finishes at a singularity, there are also processes with a nonzero amplitude in which the universe starts and finishes in the same regular configuration without ever going through a singularity. These latter processes may be pictured as a loop in the configuration space of the gravitational field. The work remains formal throughout in that no definite meaning is given to the functional integrals considered.