Anti-de Sitterian "massive" elementary systems and their Minkowskian and Newton-Hooke contraction limits

TL;DR

This work formulates massive elementary systems in AdS$_4$ by exploiting the Sp$(4,\mathbb{R})$ symmetry, identifying the classical phase space with the bounded symmetric domain $\mathcal{D}^{(3)}$ and quantizing via the discrete-series UIRs acting on Segal–Bargmann–Fock spaces. It analyzes contraction limits to the Poincaré and Newton–Hooke groups, showing that each AdS$_4$ massive system naturally decomposes into a Minkowskian massive particle plus a curvature-induced isotropic oscillator, with rest energy $E^{\rm rest}_{\rm AdS_4}=\hbar c \varkappa\,\varsigma$. The quantum treatment uses weighted Hilbert spaces and reproducing kernels, deriving explicit actions of the generators and confirming that the contraction limits recover the familiar Poincaré and Newton–Hooke UIRs while preserving coherent phase-space structure. The results suggest a matter–vibration duality in AdS$_4$ that could have implications for energy partition and possibly dark matter phenomena, grounded in a covariant, quaternionic, and Kähler-geometric framework.

Abstract

We elaborate the definition and properties of "massive" elementary systems in the $(1+3)$-dimensional Anti-de Sitter (AdS$_4$) spacetime, on both classical and quantum levels. We fully exploit the symmetry group {isomorphic to} Sp$(4,R)$, that is, 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 group of motions in AdS$_4$ spacetime. In particular, we discuss that the group coset Sp$(4,\mathbb R)/\mathrm{S}\big(\mathrm{U}(1)\times \mathrm{SU}(2)\big)$, 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 AdS$_4$ 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$(4,\mathbb R)$ 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 AdS$_4$ spacetime, as each being a combination of a Minkowskian-like elementary system {with positive proper mass}, with an isotropic harmonic oscillator arising from the AdS$_4$ 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.