Plane waves, harmonic analysis in de Sitter and anti de Sitter Quantum Field Theory and the spectral condition

We review the role of the spectral condition as a characteristic feature unifying Minkowski, de Sitter and anti de Sitter Quantum Field Theory. In this context, we highlight the role of an important class of plane waves which are either de Sitter or anti de Sitter covariant and are compatible with the relevant analyticity domains linked to the spectral condition(s). We show again how to expand the two-point functions and propagators in terms of them and some of the advantages of doing so rather than using special coordinate systems and separated variables.