Spherical formulation for diagrammatic evaluations on a manifold with boundary

Abstract

The mathematical formalism necessary for the diagrammatic evaluation of quantum corrections to a conformally invariant field theory for a self-interacting scalar field on a curved manifold with boundary is considered. The evaluation of quantum corrections to the effective action past one-loop necessitates diagrammatic techniques. Diagrammatic evaluations and higher loop-order renormalization can be best accomplished on a Riemannian manifold of constant curvature accommodating a boundary of constant extrinsic curvature. In such a context, the stated evaluations can be accomplished through a consistent interpretation of the Feynman rules within the spherical formulation of the theory which the method of images allows. To this effect, the mathematical consequences of such an interpretation are analysed and the spherical formulation of the Feynman rules on the bounded manifold is, as a result, developed.