Anti-de Sitterian "massive" elementary systems and their Minkowskian and
Newtonian limits
- URL: http://arxiv.org/abs/2307.06690v1
- Date: Thu, 13 Jul 2023 11:22:58 GMT
- Title: Anti-de Sitterian "massive" elementary systems and their Minkowskian and
Newtonian limits
- Authors: Mohammad Enayati, Jean-Pierre Gazeau, Mariano A. del Olmo, Hamed
Pejhan
- Abstract summary: ''Massive" elementary systems are defined in the (1+3)-dimensional Anti-Hook-de Sitter (AdS$_4$) spacetime.
We exploit the symmetry group Sp$(4,mathbb R)$, recognized as the relativity/kinematical group of motions in AdS$_4$ spacetime.
We unveil the dual nature of ''massive" elementary systems living in AdS$_4$ spacetime.
- Score: 0.3441021278275805
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- 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 Sp$(4,\mathbb R)$, 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$(4,\mathbb R)$/S(U(1)x 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 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$(4,\mathbb R)$ 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 (Poincar\'{e}) 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 massive
elementary system 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 r\^{o}le in the explanation of the current existence of dark matter.
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