Anti-de Sitter, plane waves and quantum field theory

We present a new plane-wave expansion of the Wightman functions of anti de Sitter scalar fields and showcase its conceptual and technical importance in AdS quantum field theory. We deduce from it a new integral representation of the Feynman propagator which helps in clarifying the relation between AdS Euclidean and Lorentzian Feynman diagrams in concrete examples. The plane-wave expansion makes it possible also to demonstrate numerous new nontrivial formulas for Bessel and Legendre functions, and we provide two examples.