The de Sitter group and its representations: a window on the notion of de Sitterian elementary systems

We review the construction of ("free") elementary systems in de Sitter (dS) spacetime, in the Wigner sense, as associated with unitary irreducible representations (UIR's) of the dS (relativity) group. This study emphasizes the conceptual issues arising in the formulation of such systems and discusses known results in a mathematically rigorous way. Particular attention is paid to: "smooth" transition from classical to quantum theory; physical content under vanishing curvature, from the point of view of a local ("tangent") Minkowskian observer; and thermal interpretation (on the quantum level), in the sense of the Gibbons-Hawking temperature. We review three decompositions of the dS group physically relevant for the description of dS spacetime and classical phase spaces of elementary systems living on it. We review the construction of (projective) dS UIR's issued from these group decompositions. (Projective) Hilbert spaces carrying the UIR's (in some restricted sense) identify quantum ("one-particle") states spaces of dS elementary systems. Adopting a well-established Fock procedure, based on the Wightman-G\"{a}rding axioms and on analyticity requirements in the complexified Riemannian manifold, we proceed with a consistent quantum field theory (QFT) formulation of elementary systems in dS spacetime. This dS QFT formulation closely parallels the corresponding Minkowskian one, while the usual spectral condition is replaced by a certain geometric Kubo-Martin-Schwinger (KMS) condition equivalent to a precise thermal manifestation of the associated "vacuum" states.