Formulae for ylm(r1×r2)

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

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“…Note the following formula relating the holomorphic solid spherical to the SU(2) UIR matrix elements [67]:…”

Section: Holomorphic Solid Spherical Harmonicsmentioning

confidence: 99%