The Spectral Condition, Plane Waves, and Harmonic Analysis in de Sitter and Anti-de Sitter Quantum Field Theories

We review the role of the spectral condition as a characteristic of Minkowski, de Sitter, and anti-de Sitter quantum field theories. We also discuss the role of plane waves which are compatible with the relevant analyticity domains linked to the spectral condition(s) and discuss harmonic analysis in terms of them.