Perturbative Evaluation of Interacting Scalar Fields on a Curved Manifold with Boundary

TL;DR

This paper performs a perturbative quantum analysis of a scalar field on a curved manifold with boundary, revealing the need for non-invariant counterterms and analyzing backreaction effects in the renormalisation process.

Contribution

It provides a detailed second-order renormalisation of scalar fields on curved manifolds with boundary, including explicit higher-loop calculations and the identification of necessary non-invariant counterterms.

Findings

Renormalisation requires non-invariant counterterms involving curvature and boundary terms.

Explicit third-loop calculations demonstrate the complexity of quantum corrections.

Backreaction effects significantly influence the renormalisation procedure.

Abstract

The effects of quantum corrections to a conformally invariant scalar field theory on a curved manifold of positive constant curvature with boundary are considered in the context of a renormalisation procedure. The renormalisation of the theory to second order in the scalar self-coupling pursued herein involves explicit calculations of up to third loop-order and reveals that, in addition to the renormalisation of the scalar self-coupling and scalar field, the removal of all divergences necessitates the introduction of conformally non-invariant counterterms proportional to $ R\Phi^2$ and $ K\Phi^2$ in the bare scalar action as well as counterterms proportional to $ RK^2$, $ R^2$ and $ RK$ in the gravitational action. The substantial backreaction effects and their relevance to the renormalisation procedure are analysed.