Covariant coordinate space methods for calculations in the quantum theory of gravity

The techniques developed to perform calculations in quantum gravidynamics must respect the general covariance of Einstein's theory of gravity. This immediately implies that coordinate space, rather than momentum space, techniques be used. In this dissertation, coordinate-space calculations of the single closed loop counter terms necessary to cancel the divergences in the action for fields on a fixed curved background are presented. Using methods developed by Schwinger and DeWitt, quartic, quadratic, and logarithmic divergences are isolated. These methods are shown to violate conformal invariance, indicating that an alternative approach is needed. Finally, using a covariant point splitting method, the vacuum expectation valve of the conformal scalar stress tensor is found. The divergent terms in the stress tensor are tracefree in the conformal case. This fact leads to the speculation that regularization methods which preserve conformal invariance in the action may exist. An anomalous finite term is found that is related to terms in an asymptotic expansion of the Feynman Green's function in inverse powers of the mass. Since this type of expansion is invalid in a massless case, a better treatment involving a cutoff mass is presented