Quantum field theory on manifolds with a boundary

We discuss quantum theory of fields Φ defined on a (d + 1)-dimensional manifold M with a boundary B. The free action W0(Φ) which is a bilinear form in Φ defines the Gaussian measure with a covariance (Green function) G. We discuss a relation between the quantum field theory with a fixed boundary condition Φ and the theory defined by the Green function G. It is shown that the latter results by an average over Φ 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 those on anti-de Sitter space