The de Sitter (dS) Group and its Representations

Abstract

We elaborate the definition and properties of "massive" elementary systems in the (1 + 3)dimensional Anti-de Sitter (AdS4) spacetime, on both classical and quantum levels. We fully exploit the symmetry group Sp(4, R), that is, the two-fold covering of SO0(2, 3) (Sp(4, R) ∼ SO0(2, 3) × Z2), recognized as the relativity/kinematical group of motions in AdS4 spacetime. In particular, we discuss that the group coset Sp(4, R)/S U(1) × SU(2) , as one of the Cartan classical domains, can be interpreted as a phase space for the set of free motions of a test massive particle on AdS4 spacetime; technically, in order to facilitate the computations, the whole process is carried out in terms of complex quaternions. The (projective) unitary irreducible representations (UIRs) of the Sp(4, R) group, describing the quantum version of such motions, are found in the discrete series of the Sp(4, R) UIRs. We also describe the null-curvature (Poincaré) and non-relativistic (Newton-Hooke) contraction limits of such systems, on both classical and quantum levels. On this basis, we unveil the dual nature of "massive" elementary systems living in AdS4 spacetime, as each being a combination of a Minkowskian-like massive elementary system with an isotropic harmonic oscillator arising from the AdS4 curvature and viewed as a Newton-Hooke elementary system. This matter-vibration duality will take its whole importance in the quantum regime (in the context of the validity of the equipartition theorem) in view of its possible rôle in the explanation of the current existence of dark matter. Contents

Asymptotic States and S-Matrix Operator in de Sitter Ambient Space Formalism

“…In Minkowski spacetime, the rest energy E rest of an elementary system is exactly its proper mass m (c = 1). These two notions decouple in dS space-time [4,11,12]. In addition, the notion of rest energy E rest dS in dS space is ill-defined due to the ambiguity in the time definition.…”

“…For the arbitrary elements of the discrete series, the scalar case, Π ± p,0 , with p = 1, 2, • • • , and the other cases, Π ± p,q , with p = 1 2 , 1, 3 2 , 2, • • • , q = p, p − 1, • • • , 1 or 1 2 , the Garidi mass reads m = h Rc [(p − q)(p + q + 1)] 1/2 , whereas (2) has no meaning for these UIRs of the dS group, particularly for the "massless minimally coupled" (mmc) scalar field representation Π ± p,0 for which ν = i3/2. All details can be found in [4]. In the sequel, we return to the atomic units h = 1 = c and the Hubble notation H = c/R = 1/R, and replace the discrete series parameter p with n.…”

“…As we mention in the introduction, we restrict our analysis to the scalar fields, since introducing fields with spin requires extra technicalities that are not fundamental to our purpose. Nevertheless, we note that the scalar case can be generalized to the various massive or massless fields with nonzero spin, corresponding to various (unitary irreducible) representations of the dS group [4,15,16].…”

“…where ξ 0 > 0, ξ α = (ξ 0 , ξ 0 u u u) ∈ C + , and u u u = (u 4 , u) is a unit vector in IR 4 . Thanks to the Fourier transformation on S 3 based on the orthogonality of the set of hyperspherical harmonics, the expansion (23) yields the following integral representation [22]:…”

“…Nevertheless, owing to the maximally symmetric nature of this spacetime with the symmetry group SO 0 (1, 4), many of the dS QFT problems can be solved. The use of both complex geometry, developed by Bros and his collaborators [2,3], and group theory, using elementary systems in the Wigner sense (see [4] for a review), have proven to be powerful tools for studying dS quantum field theory in the framework of dS ambient space formalism (see [5]).…”