Solved and Unsolved Problems in the Stochastic Quantization of Gravity

Abstract

The fact that a well-defined quantum version of Einstein’s theory of General Relativity does not exist may be taken as an indication that it is the low-energy effective field theory of a more fundamental theory, for which, at present, string theory appears to be the most promising candidate. But even if this point of view is taken for granted, Einstein gravity is far from obsolete, and it is of interest to apply alternative quantization schemes to it. The stochastic quantization method of Parisi and Wu (1981) is particularly attractive, because, in principle, computations of physical quantities can be carried out without ever having to break gauge or general coordinate invariance. If one tries to apply the method to General Relativity, however, one soon learns that it is not viable. There are two obstacles that have to be overcome in order to make stochastic quantum gravity at least formally consistent. The first one is the indefiniteness problem, i.e. the fact that the Euclidean Einstein action is not bounded from below. We shall solve this problem by formulating a Minkowski space version of stochastic quantization. The second problem may be called the covariance problem and requires a generally covariant version of stochastic quantization for its remedy. These necessary generalizations of the original Parisi-Wu formalism are the main topics of this review. We shall see at the end that, perturbatively, the solution of problem two even provides another solution to problem one.