Mass in de Sitter and Anti-de Sitter Universes with Regard to Dark Matter

An explanation of the origin of dark matter is suggested in this work. The argument is based on symmetry considerations about the concept of mass. In Wigner’s view, the rest mass and the spin of a free elementary particle in flat space-time are the two invariants that characterize the associated unitary irreducible representation of the Poincaré group. The Poincaré group has two and only two deformations with maximal symmetry. They describe respectively the de Sitter (dS) and anti-de Sitter (AdS) kinematic symmetries. Analogously to their shared flat space-time limit, two invariants, spin and energy scale for de Sitter and rest energy for anti-de Sitter, characterize the unitary irreducible representation associated with dS and AdS elementary systems, respectively. While the dS energy scale is a simple deformation of the Poincaré rest energy and so has a purely mass nature, AdS rest energy is the sum of a purely mass component and a kind of zero-point energy derived from the curvature. An analysis based on recent estimates on the chemical freeze-out temperature marking in Early Universe the phase transition quark–gluon plasma epoch to the hadron epoch supports the guess that dark matter energy might originate from an effective AdS curvature energy.

Mass in de Sitter and Anti-de-Sitter Universes with Regard to Dark Matter

An explanation of the origin of dark matter is suggested in this work. The argument is based on symmetry considerations about the concept of mass. In the Wigner's view, the rest mass and the spin of a free elementary particle in flat space-time are the two invariants that characterize the associated unitary irreducible representation of the Poincar\'e group. The Poincar\'e group has two and only two deformations with maximal symmetry. They describe respectively the de Sitter (dS) and Anti de Sitter (AdS) kinematic symmetries. Analogously to their shared flat space-time limit, two invariants, spin and energy scale for de Sitter and rest energy for Anti de Sitter, characterize the unitary irreducible representation associated with dS and AdS elementary systems. While the dS energy scale is a simple deformation of the Poincaré rest energy and so has a purely mass nature, AdS rest energy is the sum of a purely mass component and a kind of zero-point energy derived from the curvature. An analysis based on recent estimates on the chemical freeze-out temperature marking in Early Universe the phase transition quark-gluon plasma epoch to the hadron epoch supports the guess that dark matter energy might originate from an effective AdS curvature energy.

Spherical HarmonicsYlm(θ,ϕ): Positive and Negative Integer Representations ofsu(1,1)forl-mandl+m

The azimuthal and magnetic quantum numbers of spherical harmonicsYlm(θ,ϕ)describe quantization corresponding to the magnitude andz-component of angular momentum operator in the framework of realization ofsu(2)Lie algebra symmetry. The azimuthal quantum numberlallocates to itself an additional ladder symmetry by the operators which are written in terms ofl. Here, it is shown that simultaneous realization of both symmetries inherits the positive and negative(l-m)- and(l+m)-integer discrete irreducible representations forsu(1,1)Lie algebra via the spherical harmonics on the sphere as a compact manifold. So, in addition to realizing the unitary irreducible representation ofsu(2)compact Lie algebra via theYlm(θ,ϕ)’s for a givenl, we can also representsu(1,1)noncompact Lie algebra by spherical harmonics for given values ofl-mandl+m.

Berezin quantization on generalized flag manifolds

Let $M=G/H$ be a generalized flag manifold where $G$ is a compact, connected, simply-connected Lie group with Lie algebra $\mathfrak{g}$ and $H$ is the centralizer of a torus. Let $\pi$ be a unitary irreducible representation of $G$ which is holomorphically induced from a character of $H$. Using a complex parametrization of a dense open subset of $M$, we realize $\pi$ on a Hilbert space of holomorphic functions. We give explicit expressions for the differential $d\pi$ of $\pi$ and for the Berezin symbols of $\pi (g)$ ($g\in G$) and $d\pi (X)$ ($X\in \mathfrak{g}$). In particular, we recover some results of S. Berceanu and we partially generalize a result of K. H. Neeb.

REALIZATION OF MINIMAL C*-DYNAMICAL SYSTEMS IN TERMS OF CUNTZ–PIMSNER ALGEBRAS

In the present article, we provide several constructions of C*-dynamical systems [Formula: see text] with a compact group [Formula: see text] in terms of Cuntz–Pimsner algebras. These systems have a minimal relative commutant of the fixed-point algebra [Formula: see text] in [Formula: see text], i.e. [Formula: see text], where [Formula: see text] is the center of [Formula: see text], which is assumed to be non-trivial. In addition, we show in our models that the group action [Formula: see text] has full spectrum, i.e. any unitary irreducible representation of [Formula: see text] is carried by a [Formula: see text]-invariant Hilbert space within [Formula: see text]. First, we give several constructions of minimal C*-dynamical systems in terms of a single Cuntz–Pimsner algebra [Formula: see text] associated to a suitable [Formula: see text]-bimodule ℌ. These examples are labelled by the action of a discrete Abelian group ℭ (which we call the chain group) on [Formula: see text] and by the choice of a suitable class of finite dimensional representations of [Formula: see text]. Second, we present a more elaborate contruction, where now the C*-algebra [Formula: see text] is generated by a family of Cuntz–Pimsner algebras. Here, the product of the elements in different algebras is twisted by the chain group action. We specify the various constructions of C*-dynamical systems for the group [Formula: see text], N ≥ 2.

FINE DISINTEGRATION OF THE LEFT REGULAR REPRESENTATION

Let Hn be the (2n + 1)-dimensional Heisenberg group. We decompose L2(Hn) as the closure of a direct sum of infinitely many left translation invariant eigenspaces (for certain systems of partial differential equations). The restriction of the left regular representation to each one of these eigenspaces disintegrates into a direct integral of unitary irreducible representations, such that each infinite dimensional unitary irreducible representation appears with multiplicity 0 or 1 in this disintegration.