Instantons in quantum gravity

The material contained in this thesis is concerned with the functional integral approach to the quantum theory of gravity. It seems to be necessary to work with metrics of positive definite signature (Euclidean metrics) and then analytically continue the result back to the Lorentzian regime. The dominant contributions to the functional integral come from metrics which are stationary points of the action, i.e. classical solutions of the Euclideanized Einstein equations. These are known as Gravitational Instantons. Boundary conditions have to be placed upon the metrics included in the functional integral, and these are determined by the physical problem being considered. Three types of boundary condition have arisen in this context, corresponding to (i) zero temperature physics, and the calculation of particle scattering amplitudes, (ii) finite temperature effects, such as black hole radiance, and (iii) the study of the structure of the gravitational vacuum on Planck length scales. Instantons in the first category are asymptotically flat in all four directions, those in the second are asymptotically flat in three directions and periodic in the fourth, and those which arise in studying the gravitational vacuum are compact without boundaries. Much of the thesis is concerned with considering these various kinds of instanton, and particularly with the effects of their non-trivial topology. One way in which this can be investigated is by means of the various topological index theorems, and these are applied to a variety of situations. Self-dual metrics seem to have particular significance in quantum gravity, and they are discussed in detail. Finally, some recent work on the calculation of the propagation of particles in the gravitational vacuum is described. (author)