scholarly journals Jacobi Polynomials as su(2, 2) Unitary Irreducible Representation

2019 ◽  
pp. 267-283 ◽  
Author(s):  
Enrico Celeghini ◽  
Mariano A. del Olmo ◽  
Miguel A. Velasco
Keyword(s):  
Irreducible Representation ◽  
Jacobi Polynomials ◽  
Unitary Irreducible Representation

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

Universe ◽  
2020 ◽  
Vol 6 (5) ◽  
pp. 66 ◽  
Author(s):  
Jean-Pierre Gazeau
Keyword(s):  
Dark Matter ◽  
Irreducible Representation ◽  
Flat Space ◽  
Space Time ◽  
Energy Scale ◽  
Unitary Irreducible Representation ◽  
Poincaré Group ◽  
De Sitter ◽  
Rest Energy ◽  
Poincare Group

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.

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

2016 ◽  
Vol 2016 ◽  
pp. 1-7
Author(s):  
H. Fakhri
Keyword(s):  
Angular Momentum ◽  
Irreducible Representation ◽  
Lie Algebra ◽  
Spherical Harmonics ◽  
Compact Manifold ◽  
Unitary Irreducible Representation ◽  
Irreducible Representations ◽  
Angular Momentum Operator ◽  
Quantum Numbers ◽  
Azimuthal Quantum Number

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.

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

2020 ◽  
Author(s):  
Jean-Pierre Gazeau
Keyword(s):  
Dark Matter ◽  
Irreducible Representation ◽  
Rest Mass ◽  
Flat Space ◽  
Space Time ◽  
Energy Scale ◽  
Unitary Irreducible Representation ◽  
De Sitter ◽  
Rest Energy ◽  
Point Energy

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.

Primitivity in representations of polycyclic groups

1980 ◽  
Vol 88 (1) ◽  
pp. 15-31 ◽  
Author(s):  
D. L. Harper
Keyword(s):  
Finite Group ◽  
Irreducible Representation ◽  
Nilpotent Group ◽  
Infinite Dimension ◽  
Finitely Generated ◽  
Polycyclic Groups

In an earlier paper (5) we showed that a finitely generated nilpotent group which is not abelian-by-finite has a primitive irreducible representation of infinite dimension over any non-absolute field. Here we are concerned primarily with the converse question: Suppose that G is a polycyclic-by-finite group with such a representation, then what can be said about G?

scholarly journals New Specific and General Linearization Formulas of Some Classes of Jacobi Polynomials

Mathematics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 74
Author(s):  
Waleed Mohamed Abd-Elhameed ◽  
Afnan Ali
Keyword(s):  
Chebyshev Polynomials ◽  
Hypergeometric Functions ◽  
Jacobi Polynomials ◽  
Algebraic Computation ◽  
Ultraspherical Polynomials ◽  
Current Article ◽  
Linearization Coefficients

The main purpose of the current article is to develop new specific and general linearization formulas of some classes of Jacobi polynomials. The basic idea behind the derivation of these formulas is based on reducing the linearization coefficients which are represented in terms of the Kampé de Fériet function for some particular choices of the involved parameters. In some cases, the required reduction is performed with the aid of some standard reduction formulas for certain hypergeometric functions of unit argument, while, in other cases, the reduction cannot be done via standard formulas, so we resort to certain symbolic algebraic computation, and specifically the algorithms of Zeilberger, Petkovsek, and van Hoeij. Some new linearization formulas of ultraspherical polynomials and third-and fourth-kinds Chebyshev polynomials are established.

scholarly journals Heat kernels of the discrete Laguerre operators

2021 ◽  
Vol 111 (2) ◽  
Author(s):  
Aleksey Kostenko
Keyword(s):  
Hardy Inequality ◽  
Jacobi Polynomials ◽  
Heat Kernels ◽  
The Other ◽  
Heat Semigroup ◽  
Stochastic Completeness ◽  
Markovian Semigroup ◽  
Other Hand ◽  
The One

AbstractFor the discrete Laguerre operators we compute explicitly the corresponding heat kernels by expressing them with the help of Jacobi polynomials. This enables us to show that the heat semigroup is ultracontractive and to compute the corresponding norms. On the one hand, this helps us to answer basic questions (recurrence, stochastic completeness) regarding the associated Markovian semigroup. On the other hand, we prove the analogs of the Cwiekel–Lieb–Rosenblum and the Bargmann estimates for perturbations of the Laguerre operators, as well as the optimal Hardy inequality.

scholarly journals New Approaches to the General Linearization Problem of Jacobi Polynomials Based on Moments and Connection Formulas

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1573
Author(s):  
Waleed Mohamed Abd-Elhameed ◽  
Badah Mohamed Badah
Keyword(s):  
Differential Equation ◽  
Hypergeometric Functions ◽  
Jacobi Polynomials ◽  
Tau Method ◽  
Riccati Differential Equation ◽  
Shifted Jacobi Polynomials ◽  
Linearization Problem ◽  
Linearization Coefficients ◽  
New Formula ◽  
Connection Formulas

This article deals with the general linearization problem of Jacobi polynomials. We provide two approaches for finding closed analytical forms of the linearization coefficients of these polynomials. The first approach is built on establishing a new formula in which the moments of the shifted Jacobi polynomials are expressed in terms of other shifted Jacobi polynomials. The derived moments formula involves a hypergeometric function of the type 4F3(1), which cannot be summed in general, but for special choices of the involved parameters, it can be summed. The reduced moments formulas lead to establishing new linearization formulas of certain parameters of Jacobi polynomials. Another approach for obtaining other linearization formulas of some Jacobi polynomials depends on making use of the connection formulas between two different Jacobi polynomials. In the two suggested approaches, we utilize some standard reduction formulas for certain hypergeometric functions of the unit argument such as Watson’s and Chu-Vandermonde identities. Furthermore, some symbolic algebraic computations such as the algorithms of Zeilberger, Petkovsek and van Hoeij may be utilized for the same purpose. As an application of some of the derived linearization formulas, we propose a numerical algorithm to solve the non-linear Riccati differential equation based on the application of the spectral tau method.

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