Perturbative Evaluation of the Zero-Point function for Self-Interacting Scalar Field on a Manifold with Boundary

TL;DR

This paper investigates quantum corrections to the gravitational action of a self-interacting scalar field on a curved manifold with boundary, highlighting the effects of topology and the need for combined volume and surface renormalization.

Contribution

It provides a third-loop perturbative analysis of the zero-point function on manifolds with boundary, emphasizing the impact of topology and boundary conditions on quantum vacuum processes.

Findings

Topology influences vacuum processes and quantum behavior.

Surface divergences require combined volume and surface renormalization.

First surface divergence is explicitly evaluated.

Abstract

The character of quantum corrections to the gravitational action of a conformally invariant field theory for a self-interacting scalar field on a manifold with boundary is considered at third loop-order in the perturbative expansion of the zero-point function. 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. The associated spherical formulation for diagramatic evaluations reveals a non-trivial effect which the topology of the manifold has on the vacuum processes and which ultimately dissociates the dynamical behaviour of the quantised field from its behaviour in the absence of a boundary. The first surface divergence is evaluated and the necessity for simultaneous renormalisation of volume and surface divergences is shown.