Topics in the Semiclassical Quantization of Gravitation

We discuss three problems in which general coordinate covariance and quantum mechanics play fundamental roles. We formulate a functional approach to scalar quantum field theory in n+1 dimensional de Sitter spacetime and solve the functional Schrodinger equation for the conformally and minimally coupled scalar fields in both the k = 0 and k = 1 gauges. We show that there is a natural initial condition, the requirement that the field energy remain finite as the scale factor a becomes small, which specifies a unique, time-dependent, de Sitter vacuum state. This initial condition is closely related to Hawking's prescription of including in the functional integral only those field configurations which are regular on the Euclidean section. The Green's functions constructed using this initial condition are explicitly shown to be the analytic continuation of those derived using the Euclidean path integral formalism and the regularity (boundary) condition. These Green's functions are used to study the Hawking effect and the restoration of continuous symmetries. In particular we study the restoration of a broken O(2) symmetry of a (PHI)('4) theory. We argue that spontaneously broken continuous symmetries are always dynamically restored in de Sitter spacetime. We discuss the canonical quantization of gravitation in the vielbein formalism and derive the Harrison-Zeldovich spectrum by perturbatively solving the Wheeler-DeWitt equations for an inflating universe coupled to a scalar field in 2+1 and 3+1 dimensions. We present a gauge invariant action which describes the propagation of the superstring in curved superspace in the presence of background super Yang-Mills fields. We show that this action possesses the local fermionic world -sheet symmetry needed for a consistent coupling of the string to background fields. Some other aspects of the superspace non-linear (sigma)-model described by this action are also discussed. We study the normal coordinate expansion in superspace and show that it may be used to generate the (THETA)-expansion of a scalar superfield Lagrangian.