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

Abstract

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-Gärding axiomatic 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. We end our study by reviewing a consistent and univocal definition of mass in dS relativity. This definition, presented in terms of invariant parameters characterizing the dS UIR's, accurately gives sense to terms like "massive" and "massless" fields in dS relativity according to their Minkowskian counterparts, yielded by the group contraction procedures. Contents

The (partially) massless spin-3/2 and spin-5/2 fields in de Sitter spacetime as unitary and non-unitary representations of the de Sitter algebra

“…Then, for N = 4, we will identify the UIR's of spin(4, 1) that correspond to the unitary strictly massless spin-3/2 field and to the unitary strictly and partially massless spin-5/2 fields on dS 4 (these UIR's have been also identified in Ref. [37]). For the sake of completeness, we also identify the UIR's of spin(N, 1) (N ≥ 3) that correspond to spin-3/2 and spin-5/2 fields with real mass parameter M on dS N .…”

Regularization of electromagnetic field for self-force problem in de Sitter spacetime

“…Causal structure of de Sitter space is induced by that in 'host' Minkowski spacetime of five dimensions [23,§ II.7]. The interval between two points, x and x ′ , in dS 4 is nothing but the projection of interval in five-dimensional flat spacetime on the hyperboloid (2.1):…”

“…Topologically, de Sitter space is the direct product R × S 3 where R is time axis and S 3 is three-dimensional sphere [23]. The non-trivial topology makes electrodynamics very specific.…”

Particles of a de Sitter Universe

“…Both of these later references compute characters of the representations and compare the separate cases of unitary irreducible representations in the presence of negative and positive cosmological constants. Another recent review which compares quantum field theory in the cases of zero and positive cosmological constants is that of [14] where group contractions and concepts of being massive and massless are also considered along other points. Our goal in this review is to draw attention to some recent directions in the discussion of de Sitter representations and their recognition in physical setups in a focused manner.…”

“…We refer to this field as the massless field; however, the concept of masslessness on de Sitter is far more involved beyond the case of zero mass scalar. The review [14] involves a better summary on the concept of masslessness.…”