Non-minimal coupling, boundary terms and renormalization of the Einstein-Hilbert action

TL;DR

This paper investigates how to maintain the balance between volume and boundary terms in the gravitational action when matter is non-minimally coupled, ensuring proper renormalization of quantum corrections and boundary divergences.

Contribution

It introduces a boundary term for non-minimally coupled matter that preserves the variational principle and ensures consistent renormalization of the effective action.

Findings

Boundary term modifies one-loop quantum corrections.

Boundary UV divergences are renormalized with volume divergences.

Results apply to arbitrary non-minimally coupled matter.

Abstract

A consistent variational procedure applied to the gravitational action requires according to Gibbons and Hawking a certain balance between the volume and boundary parts of the action. We consider the problem of preserving this balance in the quantum effective action for the matter non-minimally coupled to metric. It is shown that one has to add a special boundary term to the matter action analogous to the Gibbons-Hawking one. This boundary term modifies the one-loop quantum corrections to give a correct balance for the effective action as well. This means that the boundary UV divergences do not require independent renormalization and are automatically renormalized simultaneously with their volume part. This result is derived for arbitrary non-minimally coupled matter. The example of 2D Maxwell field is considered in much detail. The relevance of the results obtained to the problem of the renormalization of the black hole entropy is discussed.