Quantum field theory on manifolds with a boundary

TL;DR

This paper explores quantum field theory on manifolds with boundaries, analyzing boundary conditions, Green functions, and the regularity of boundary fields, with applications to (anti)de Sitter space.

Contribution

It establishes a relation between boundary conditions and Green functions in quantum field theory on manifolds with boundaries, including explicit examples in (anti)de Sitter space.

Findings

Quantum fields on the boundary are more regular than in the bulk.

The Green function approach relates boundary conditions to quantum field measures.

Application to (anti)de Sitter space illustrates the theory.

Abstract

We discuss quantum theory of fields \phi defined on (d+1)-dimensional manifold {\cal M} with a boundary {\cal B}. The free action W_{0}(\phi) which is a bilinear form in \phi defines the Gaussian measure with a covariance (Green function) {\cal G}. We discuss a relation between the quantum field theory with a fixed boundary condition \Phi and the theory defined by the Green function {\cal G}. It is shown that the latter results by an average over \Phi of the first. The QFT in (anti)de Sitter space is treated as an example. It is shown that quantum fields on the boundary are more regular than the ones on (anti) de Sitter space.