Quantum field theory on manifolds with a boundary

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