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

TL;DR

This work shows that for non-minimally coupled matter, a boundary term analogous to the Gibbons–Hawking piece is required to achieve a well-defined variational problem and to preserve the volume–boundary balance in the quantum effective action. By deriving the appropriate boundary term for arbitrary non-minimal couplings and computing the heat kernel coefficient $a_1$, the authors demonstrate that the linear-curvature part of the quantum action naturally adopts an Einstein–Hilbert–like form, with both bulk and boundary UV divergences renormalized together by the gravitational constant $G$. The 2D Maxwell example illustrates the role of boundary conditions in boundary contributions, and the renormalization of black hole entropy follows from the same boundary-bulk balance, with explicit expressions for scalar and vector cases. Overall, the results establish a general mechanism linking boundary terms, renormalization, and black hole entropy for a broad class of non-minimally coupled matter. The approach differs from prior conical-method analyses and highlights the universality of the boundary–bulk renormalization in quantum gravity contexts.

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.