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We elaborate the definition and properties of ''massive" elementary systemsin the (1+3)-dimensional Anti-de Sitter (AdS$_4$) spacetime, on both classicaland quantum levels. We fully exploit the symmetry group Sp$(4,\mathbb R)$, thatis, the two-fold covering of SO$_0(2,3)$ (Sp$(4,\mathbb R) \sim$SO$_0(2,3)\times \mathbb Z_2$), recognized as the relativity/kinematical groupof motions in AdS$_4$ spacetime. In particular, we discuss that the group cosetSp$(4,\mathbb R)$/S(U(1)x SU(2)), as one of the Cartan classical domains, canbe interpreted as a phase space for the set of free motions of a test massiveparticle on AdS$_4$ spacetime; technically, in order to facilitate thecomputations, the whole process is carried out in terms of complex quaternions.The (projective) unitary irreducible representations (UIRs) of theSp$(4,\mathbb R)$ group, describing the quantum version of such motions, arefound in the discrete series of the Sp$(4,\mathbb R)$ UIRs. We also describethe null-curvature (Poincar\'{e}) and non-relativistic (Newton-Hooke)contraction limits of such systems, on both classical and quantum levels. Onthis basis, we unveil the dual nature of ''massive" elementary systems livingin AdS$_4$ spacetime, as each being a combination of a Minkowskian-like massiveelementary system with an isotropic harmonic oscillator arising from theAdS$_4$ curvature and viewed as a Newton-Hooke elementary system. Thismatter-vibration duality will take its whole importance in the quantum regime(in the context of the validity of the equipartition theorem) in view of itspossible r\^{o}le in the explanation of the current existence of dark matter.

Anti-de Sitterian "massive" elementary systems and their Minkowskian and Newtonian limits