Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Physics A Mathematical and General

Q and P representatives for spherical tensors

Robert Gilmore

Published Jul 1, 1976• 24 citations
Publisher

Abstract

The Q and P-representatives for the irreducible spherical tensor operators Y M L (J) are proportional to the corresponding spherical harmonics Y M L ( theta , phi ). In the coherent state basis associated with the (2j+1)-dimensional unitary irreducible representation of SU(2), the proportionality factors are (2j)!/(2j-L)!2 L and (2j+1+L)!/(2j+1)!2 L , respectively.

DOI
10.1088/0305-4470/9/7/001

Related papers

Coherent states for the harmonic series irreducible representations of U(p,q). I. Unitary operator coherent states

C. Quesne

Unitary operator coherent states, as defined by Klauder (1963), Perelomov (1972) and Gilmore (1974), are considered for the harmonic series irreducible representations of U(p,q). Various properties of these states are reviewed, including their reproducing kernel and measure explicit forms, as well as the representation they lead to for the U(p,q) generators. Starting from a contraction of the su(p,q) Lie algebra, both Dyson and Holstein-Primakoff boson realisations of u(p,q) are obtained in terms of pq pairs of boson creation, annihilation operators and of the generators of a U(p)*U(q) intrinsic group. Such boson realisations are applied to determine the matrix elements of the U(p,q) generators between discrete bases classified according to the chain U(p,q) contains/implies U(p)*U(q). Finally, the isomorphism between so(4,2) and su(2,2) is employed to derive the corresponding properties of the SO(4,2) unitary operator coherent states for application in atomic physics.

Journal of Physics A Mathematical and General•Mar 11, 1987•2 citations

Coherency of su(1,1)-Barut–Girardello type and entanglement for spherical harmonics

Hossein Fakhri, Alireza Dehghani

Barut–Girardello coherent states corresponding to the (l−m)- and (l+m)-integer discrete irreducible representations of su(1,1) Lie algebra are calculated by the spherical harmonics Ylm(θ,ϕ). Their explicit compact forms and also, to realize the resolution of the identity, their corresponding positive definite measures on the complex plane are obtained in terms of the known functions. It is also shown that coherent states of both positive and negative representations separately lead us to construct new representation bases for su(1,1) Lie algebra. Then, it is shown that the su(1,1)-Barut–Girardello coherent states corresponding to two particles containing the spatial parity symmetries of a bipartite quantum system can be entangled in ten different ways.

Journal of Mathematical Physics•May 1, 2009•21 citations

Relations between Cartesian and spherical components of irreducible Cartesian tensors

J -M Normand, Jacques Raynal

Explicit formulae that relate Cartesian and spheric components of irreducible Cartesian tensors are derived. For the description of spin-s particles, explicit formulae for the Cartesian tensors and the spherical tensors are given in terms of symmetrised products of spin operators.

Journal of Physics A Mathematical and General•May 1, 1982•22 citations

Addition theorems for spin spherical harmonics. I Preliminaries

Antonio O. Bouzas

We develop a systematic approach to deriving addition theorems for, and some other bilocal sums of, spin spherical harmonics. In this first part we establish some necessary technical results. We discuss the factorization of orbital and spin degrees of freedom in certain products of Clebsch-Gordan coefficients, and obtain general explicit results for the matrix elements in configuration space of tensor products of arbitrary rank of the position and angular-momentum operators. These results are the basis of the addition theorems for spin spherical harmonics obtained in part II.

•Jan 1, 2013•4 citations

Addition theorems for spin spherical harmonics: I. Preliminaries

Antonio O Bouzas

We develop a systematic approach to deriving addition theorems for, and some other bilocal sums of, spin spherical harmonics. In this first part we establish some necessary technical results. We discuss the factorization of orbital and spin degrees of freedom in certain products of Clebsch–Gordan coefficients, and obtain general explicit results for the matrix elements in configuration space of tensor products of arbitrary rank of the position and angular-momentum operators. These results are the basis of the addition theorems for spin spherical harmonics obtained in part II (2011 J. Phys. A: Math. Theor. 44 165302 ).

arXiv:1103.2982•Journal of Physics A Mathematical and Theoretical•Mar 24, 2011•5 citations

Representations of the three-dimensional rotation group by the direct method

A. J. Torruella

The irreducible representations of the three-dimensional rotation group are obtained directly from the irreducible representations of its infinitesimal generators (the spin matrices), parametrized in terms of the rotation angle and the direction of the rotation axis. Expressions are given for the rotation operator exp(iψn ⋅ S) in terms of two different bases of 2j+1 elements for spin j. The results are related to the spectral decomposition of the rotation operator and expressions obtained for spin projection operators along any spatial direction for arbitrary spin.

Journal of Mathematical Physics•Aug 1, 1975•10 citations
The Q and P-representatives for the irreducible spherical tensor operators Y M L (J) are proportional to the corresponding spherical harmonics Y M L ( theta , phi ). In the coherent state basis associated with the (2j+1)-dimensional unitary irreducible representation of SU(2), the proportionality factors are (2j)!/(2j-L)!2 L and (2j+1+L)!/(2j+1)!2 L , respectively.
Unitary operator coherent states, as defined by Klauder (1963), Perelomov (1972) and Gilmore (1974), are considered for the harmonic series irreducible representations of U(p,q). Various properties of these states are reviewed, including their reproducing kernel and measure explicit forms, as well as the representation they lead to for the U(p,q) generators. Starting from a contraction of the su(p,q) Lie algebra, both Dyson and Holstein-Primakoff boson realisations of u(p,q) are obtained in terms of pq pairs of boson creation, annihilation operators and of the generators of a U(p)*U(q) intrinsic group. Such boson realisations are applied to determine the matrix elements of the U(p,q) generators between discrete bases classified according to the chain U(p,q) contains/implies U(p)*U(q). Finally, the isomorphism between so(4,2) and su(2,2) is employed to derive the corresponding properties of the SO(4,2) unitary operator coherent states for application in atomic physics.
Barut–Girardello coherent states corresponding to the (l−m)- and (l+m)-integer discrete irreducible representations of su(1,1) Lie algebra are calculated by the spherical harmonics Ylm(θ,ϕ). Their explicit compact forms and also, to realize the resolution of the identity, their corresponding positive definite measures on the complex plane are obtained in terms of the known functions. It is also shown that coherent states of both positive and negative representations separately lead us to construct new representation bases for su(1,1) Lie algebra. Then, it is shown that the su(1,1)-Barut–Girardello coherent states corresponding to two particles containing the spatial parity symmetries of a bipartite quantum system can be entangled in ten different ways.
We develop a systematic approach to deriving addition theorems for, and some other bilocal sums of, spin spherical harmonics. In this first part we establish some necessary technical results. We discuss the factorization of orbital and spin degrees of freedom in certain products of Clebsch-Gordan coefficients, and obtain general explicit results for the matrix elements in configuration space of tensor products of arbitrary rank of the position and angular-momentum operators. These results are the basis of the addition theorems for spin spherical harmonics obtained in part II.
We develop a systematic approach to deriving addition theorems for, and some other bilocal sums of, spin spherical harmonics. In this first part we establish some necessary technical results. We discuss the factorization of orbital and spin degrees of freedom in certain products of Clebsch–Gordan coefficients, and obtain general explicit results for the matrix elements in configuration space of tensor products of arbitrary rank of the position and angular-momentum operators. These results are the basis of the addition theorems for spin spherical harmonics obtained in part II (2011 J. Phys. A: Math. Theor. 44 165302 ).
The irreducible representations of the three-dimensional rotation group are obtained directly from the irreducible representations of its infinitesimal generators (the spin matrices), parametrized in terms of the rotation angle and the direction of the rotation axis. Expressions are given for the rotation operator exp(iψn ⋅ S) in terms of two different bases of 2j+1 elements for spin j. The results are related to the spectral decomposition of the rotation operator and expressions obtained for spin projection operators along any spatial direction for arbitrary spin.