Spherical Formulation for Diagramatic Evaluations on a Manifold with Boundary

TL;DR

This paper develops a spherical formulation of Feynman rules for diagrammatic quantum corrections on curved manifolds with boundary, enabling higher-loop calculations in conformal field theories.

Contribution

It introduces a novel spherical approach to diagrammatic evaluations on manifolds with boundary, facilitating multi-loop renormalization in curved space quantum field theories.

Findings

Formulation of Feynman rules on spherical manifolds with boundary

Application of the method of images in the spherical formalism

Enabling higher-loop quantum correction calculations

Abstract

The mathematical formalism necessary for the diagramatic 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 diagramatic techniques. Diagramatic evaluations and higher loop-order renormalisation 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 for which the method of images allows. To this effect, the mathematical consequences of such an interpretation are analyzed and the spherical formulation of the Feynman rules on the bounded manifold is, as a result, developed.