Quantum Field Theories on Manifolds with Curved Boundaries: Scalar Fields

TL;DR

This paper develops a perturbative framework for quantum field theories on manifolds with curved boundaries, incorporating boundary curvature effects and verifying consistency with renormalisation group and conformal invariance at critical points.

Contribution

It introduces a heat kernel expansion approach for arbitrary curved boundaries and applies it to scalar field theories, including boundary curvature effects in renormalisation and RG functions.

Findings

Renormalisation group functions agree with previous results for plane boundaries.

Boundary curvature terms influence the RG functions and critical behavior.

The local Schrödinger equation for the wave functional is consistent with the renormalisation group.

Abstract

A framework allowing for perturbative calculations to be carried out for quantum field theories with arbitrary smoothly curved boundaries is described. It is based on an expansion of the heat kernel derived earlier for arbitrary mixed Dirichlet and Neumann boundary conditions. The method is applied to a general renormalisable scalar field theory in four dimensions using dimensional regularisation to two loops and expanding about arbitrary background fields. Detailed results are also specialised to an $O(n)$ symmetric model with a single coupling constant. Extra boundary terms are introduced into the action which give rise to either Dirichlet or generalised Neumann boundary conditions for the quantum fields. For plane boundaries the resulting renormalisation group functions are in accord with earlier results but here the additional terms depending on the extrinsic curvature of the boundary are found. Various consistency relations are also checked and the implications of conformal invariance at the critical point where the $\beta$ function vanishes are also derived. The local Scr\"odinger equation for the wave functional defined by the functional integral under deformations of the boundary is also verified to two loops. Its consistency with the renormalisation group to all orders in perturbation theory is discussed.