Quantum Field Theories on Manifolds with Curved Boundaries: Scalar Fields

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 Green function for second order differential operators valid in the neighbourhood of the boundary and which is obtained from a corresponding expansion of of the associated heat kernel derived earlier for arbitrary mixed Dirichlet and Neumann boundary conditions. The first few leading terms in the expansion are sufficient to calculate all additional divergences present in a perturbative loop expansion as a consequence of the presence of the boundary. 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 β function vanishes are also derived. For a general scalar field theory, where the fields φ attain specified values φ̂ on the boundary, the local Scrödinger equation for the wave functional defined by the functional integral under deformations of the boundary is also verified to two loops. The perturbative expansion for the wave functional is defined by expansion around the solution of the classical field equations satisfying the required boundary values and the counterterms necessary to derive a finite hamiltonian operator, which includes a functional laplace operator on the fields φ̂, are found to the order considered. Consistency of the local Schrödinger equation with the renormalisation group to all orders in perturbation theory is also discussed.