Clebsch-Gordan Coefficients Research Articles

Interbasis expansions in the Zernike system

The differential equation with free boundary conditions on the unit disk that was proposed by Frits Zernike in 1934 to find Jacobi polynomial solutions (indicated as I) serves to define a classical system and a quantum system which have been found to be superintegrable. We have determined two new orthogonal polynomial solutions (indicated as II and III) that are separable and involve Legendre and Gegenbauer polynomials. Here we report on their three interbasis expansion coefficients: between the I–II and I–III bases, they are given by F23(⋯|1) polynomials that are also special su(2) Clebsch–Gordan coefficients and Hahn polynomials. Between the II–III bases, we find an expansion expressed by F34(⋯|1)’s and Racah polynomials that are related to the Wigner 6j coefficients.

A new algorithm for computing branching rules and Clebsch–Gordan coefficients of unitary representations of compact groups

A numerical algorithm that computes the decomposition of any finite-dimensional unitary reducible representation of a compact Lie group is presented. The algorithm, which does not rely on an algebraic insight into the group structure, is inspired by quantum mechanical notions. After generating two adapted states (these objects will be conveniently defined in Definition II.1) and after appropriate algebraic manipulations, the algorithm returns the block matrix structure of the representation in terms of its irreducible components. It also provides an adapted orthonormal basis. The algorithm can be used to compute the Clebsch–Gordan coefficients of the tensor product of irreducible representations of a given compact Lie group. The performance of the algorithm is tested on various examples: the decomposition of the regular representation of two finite groups and the computation of Clebsch–Gordan coefficients of two examples of tensor products of representations of SU(2).

New separated polynomial solutions to the Zernike system on the unit disk and interbasis expansion.

The differential equation proposed by Frits Zernike to obtain a basis of polynomial orthogonal solutions on the unit disk to classify wavefront aberrations in circular pupils is shown to have a set of new orthonormal solution bases involving Legendre and Gegenbauer polynomials in nonorthogonal coordinates, close to Cartesian ones. We find the overlaps between the original Zernike basis and a representative of the new set, which turn out to be Clebsch-Gordan coefficients.

POVM construction: A simple recipe with applications to symmetric states

We propose a simple method for constructing positive operator-valued measures (POVMs) using any set of matrices which form an orthonormal basis for the space of complex matrices. Considering the orthonormal set of irreducible spherical tensors, we examine the properties of the construction on the [Formula: see text]-dimensional subspace of the [Formula: see text]-dimensional Hilbert space of [Formula: see text] qubits comprising the permutationally symmetric states. Using the notion of vectorization, the constructed POVMs are interpretable as projection operators in a higher-dimensional space. We then describe a method to physically realize the constructed POVMs for symmetric states using the Clebsch–Gordan decomposition of the tensor product of irreducible representations of the rotation group. We illustrate the proposed construction on a spin-1 system, and show that it is possible to generate entangled states from separable ones.

Polarization-Dressed Correlation and Anti-Correlation Between Phase-Conjugate Four-Wave Mixing and Fourth-Order Fluorescence

We report the two- and three-mode correlation of phase-conjugate four-wave mixing and fourth-order fluorescence in Pr <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3+</sup> :Y <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> SiO <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">5</sub> crystal. The degree of correlation is controlled by changing the position of detecting time and polarization state of pumping field. The switching between correlation and anti-correlation was observed at different positions of detecting time because of the competition between four-wave mixing and fourth-order fluorescence signals. The obvious variation of correlation degree was also observed at different polarization states, which is interpreted by the different Clebsch-Gordan coefficients and the EIT subsystem induced by polarization dressing effect.

Generalized Uncertainty Principle and angular momentum

Various models of quantum gravity suggest a modification of the Heisenberg’s Uncertainty Principle, to the so-called Generalized Uncertainty Principle, between position and momentum. In this work we show how this modification influences the theory of angular momentum in Quantum Mechanics. In particular, we compute Planck scale corrections to angular momentum eigenvalues, the hydrogen atom spectrum, the Stern–Gerlach experiment and the Clebsch–Gordan coefficients. We also examine effects of the Generalized Uncertainty Principle on multi-particle systems.

Permutation-symmetric three-particle hyper-spherical harmonics based on the S3 ⊗ SO(3)rot ⊂ O(2)⊗SO(3)rot ⊂ U(3)⋊S2 ⊂ O(6) subgroup chain

We construct the three-body permutation symmetric hyperspherical harmonics to be used in the non-relativistic three-body Schrödinger equation in three spatial dimensions (3D). We label the state vectors according to the S3⊗SO(3)rot⊂O(2)⊗SO(3)rot⊂U(3)⋊S2⊂O(6) subgroup chain, where S3 is the three-body permutation group and S2 is its two element subgroup containing transposition of first two particles, O(2) is the “democracy transformation”, or “kinematic rotation” group for three particles; SO(3)rot is the 3D rotation group, and U(3),O(6) are the usual Lie groups. We discuss the good quantum numbers implied by the above chain of algebras, as well as their relation to the S3 permutation properties of the harmonics, particularly in view of the SO(3)rot⊂SU(3) degeneracy. We provide a definite, practically implementable algorithm for the calculation of harmonics with arbitrary finite integer values of the hyper angular momentum K, and show an explicit example of this construction in a specific case with degeneracy, as well as tables of K≤6 harmonics. All harmonics are expressed as homogeneous polynomials in the Jacobi vectors (λ,ρ) with coefficients given as algebraic numbers unless the “operator method” is chosen for the lifting of the SO(3)rot⊂SU(3) multiplicity and the dimension of the degenerate subspace is greater than four – in which case one must resort to numerical diagonalization; the latter condition is not met by any K≤15 harmonic, or by any L≤7 harmonic with arbitrary K. We also calculate a certain type of matrix elements (the Gaunt integrals of products of three harmonics) in two ways: 1) by explicit evaluation of integrals and 2) by reduction to known SU(3) Clebsch–Gordan coefficients. In this way we complete the calculation of the ingredients sufficient for the solution to the quantum-mechanical three-body bound state problem.

WEAK HOPF ALGEBRAS CORRESPONDING TO NON-STANDARD QUANTUM GROUPS

We construct a weak Hopf algebra <TEX>$wX_q(A_1)$</TEX> corresponding to non-standard quantum group <TEX>$X_q(A_1)$</TEX>. The PBW basis of <TEX>$wX_q(A_1)$</TEX> is described and all the highest weight modules of <TEX>$wX_q(A_1)$</TEX> are classified. Finally we give the Clebsch-Gordan decomposition of the tensor product of two highest weight modules of <TEX>$wX_q(A_1)$</TEX>.

Boosting Wigner’s nj-symbols

We study the SL(2,ℂ) Clebsch-Gordan coefficients appearing in the Lorentzian EPRL spin foam amplitudes for loop quantum gravity. We show how the amplitudes decompose into SU(2) nj− symbols at the vertices and integrals over boosts at the edges. The integrals define edge amplitudes that can be evaluated analytically using and adapting results in the literature, leading to a pure state sum model formulation. This procedure introduces virtual representations which, in a manner reminiscent of virtual momenta in Feynman amplitudes, are off-shell of the simplicity constraints present in the theory, but with the integrands that peak at the on-shell values. We point out some properties of the edge amplitudes which are helpful for numerical and analytical evaluations of spin foam amplitudes, and suggest among other things a simpler model useful for calculations of certain lowest order amplitudes. As an application, we estimate the large spin scaling behaviour of the simpler model, on a closed foam with all 4-valent edges and Euler characteristic χ, to be Nχ−5E+V/2. The paper contains a review and an extension of the results on SL(2,ℂ) Clebsch-Gordan coefficients among unitary representations of the principal series that can be useful beyond their application to quantum gravity considered here.

The Partition Formalism and New Entropic-Information Inequalities for Real Numbers on an Example of Clebsch–Gordan Coefficients

We discuss the procedure of different partitions in the finite set of N integer numbers and construct generic formulas for a bijective map of real numbers sy, where y = 1, 2,…, N, N = \( \underset{k=1}{\overset{n}{\varPi}}{X}_k, \) and Xk are positive integers, onto the set of numbers s(y(x1, x2,…, xn)). We give the functions used to present the bijective map, namely, y(x1, x2, …, xn) and xk(y) in an explicit form and call them the functions detecting the hidden correlations in the system. The idea to introduce and employ the notion of “hidden gates” for a single qudit is proposed. We obtain the entropic-information inequalities for an arbitrary finite set of real numbers and consider the inequalities for arbitrary Clebsch–Gordan coefficients as an example of the found relations for real numbers.

Generating functions for the Clebsch–Gordan coefficients

Generating functions for Clebsch–Gordan coefficients of are derived. These coefficients are expressed as limits of the dual q-Hahn polynomials. The generating functions are obtained using two different approaches respectively based on the coherent-state representation and the position representation of

Generic Representation of Y(s o(3)) Based on the Lie Algebraic Basis of s o(3)

We focus on constructing a generic representation of Y(so(3)) based on the Lie algebraic basis of so(3) basis, and further developing transition of Yangian operator \(\hat {\mathbf {Y}}\). As an application of Y(so(3)), we calculate all the matrix elements of unit vector operators \(\hat {\mathbf {n}}\) in angular momentum basis. It is also discovered that the Yangian operator \(\hat {\mathbf {Y}}\) may work in quantum vector space. In addition, some shift operators \(\hat {O}^{(\pm )}_{\mu }\) are naturally built on the basis of the representation of Y(so(3)). As an another application of Y(so(3)), we can derive the CG cofficients of two coupled angular momenta from the down-shift operator \(\hat {O}^{(-)}_{-1}\) acting on a so(3)-coupled tensor basis. This not only explores that Yangian algebras can work in quantum tensor space, but also provides a novel approach to solve CG coefficients for two coupled angular momenta.

A computer code for calculations in the algebraic collective model of the atomic nucleus

A Maple code is presented for algebraic collective model (ACM) calculations. The ACM is an algebraic version of the Bohr model of the atomic nucleus, in which all required matrix elements are derived by exploiting the model’s SU(1,1)×SO(5) dynamical group. This paper reviews the mathematical formulation of the ACM, and serves as a manual for the code.The code enables a wide range of model Hamiltonians to be analysed. This range includes essentially all Hamiltonians that are rational functions of the model’s quadrupole moments qˆM and are at most quadratic in the corresponding conjugate momenta πˆN (−2≤M,N≤2). The code makes use of expressions for matrix elements derived elsewhere and newly derived matrix elements of the operators [πˆ⊗qˆ⊗πˆ]0 and [πˆ⊗πˆ]LM. The code is made efficient by use of an analytical expression for the needed SO(5)-reduced matrix elements, and use of SO(5)⊃SO(3) Clebsch–Gordan coefficients obtained from precomputed data files provided with the code. Program summaryProgram title: ACMCatalogue identifier: AEYO_v1_0Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEYO_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 3873526No. of bytes in distributed program, including test data, etc.: 46345414Distribution format: tar.gzProgramming language: Maple 18 (or versions 17, 16, 15).Computer: Any.Operating system: Any which supports Maple; tested under Linux, Max OSX, Windows 7.RAM: 500MbClassification: 17.20.Nature of problem: The calculation of energy eigenvalues, transition rates and amplitudes of user specified Hamiltonians in the Bohr model of the atomic nucleus.Solution method: Exploit the model’s SU(1,1)×SO(5) dynamical group to calculate analytic (as far as possible) expressions for matrix elements, making use of extensive files (supplied) of SO(5)⊃SO(3) Clebsch–Gordan coefficients. Diagonalisation of the resulting matrices (once the entries are converted to floating point) is carried out using the Maple library procedure Eigenvectors. (Maple [1] makes use of the NAG [2] and CLAPACK [3] linear algebra libraries.)Additional comments: 1.The dimension of the Hilbert space that can be handled is limited only by the available computer memory and the available SO(5)⊃SO(3) Clebsch–Gordan coefficients (v1α1L1v2α2L2∥v3α3L3).2.The supplied data files provide coefficients (v1α1L1v2α2L2∥v3α3L3) for 1≤v2≤6, and contain all non-zero coefficients for v1<v3≤50 when v2∈1,3, for v1≤v3≤30 when v2∈2,4, and for v1≤v3≤25 when v2∈5,6. (Once calculated, further coefficients can be readily made available to the code without changing the code.) Thus, depending on the model Hamiltonian being analysed, the states in the Hilbert space used are limited in their seniority. For analysis of the more typical types of model Hamiltonian, only the coefficients with v2∈{1,3} are required, and therefore, with the supplied files, the seniority limit is 50. More exotic Hamiltonians having terms with seniority v2∈{2,4,5,6} would have the seniority limited to 30 or 25 accordingly.3.The code provides lower level procedures that give ready access to the Clebsch–Gordan coefficients and the SU(1, 1) and SO(5) matrix elements. These procedures are described in the manuscript and enable extensions to the code and model to be made easily.4.The accuracy to which Maple performs numerical calculations is determined by the Maple parameter Digits, which specifies the number of significant decimal digits used. The default value of 10 is more than adequate for most ACM calculations. Note, however, that if Digits is increased beyond a certain value (obtained from the Maple command evalhf(Digits), and usually 15 on modern computers) then the code can no longer take advantage of hardware mathematical operations, and is significantly slower.Documents included1.The code makes use of SO(5)⊃SO(3) Clebsch–Gordan coefficients which are supplied in zip files, and must be installed by the user.2.A Maple worksheet that gives various example calculations and tests carried out using procedures from the code is provided.3.A 162 page PDF file containing everything displayed in the worksheet (input, output and comments, and making use of colour) is also provided. !!!!! The distribution file for this program is over 46 Mbytes and therefore is not delivered directly when download or Email is requested. Instead a html file giving details of how the program can be obtained is sent. !!!!!Running time: For a fixed value of the parameter Digits, the running time depends on the dimension of the Hilbert space on which the diagonalisation is performed, and this in turn is governed by the number of eigenvalues required and the accuracy required. Note that diagonalisation is performed separately in each L-space. For typical ACM calculations (such as those carried out in [4]), the matrices being diagonalised are usually of dimension at most a few hundred, and often much smaller. On a modest personal computer, the computation for the smallest cases takes at most a few seconds. The worksheet contains a range of examples for which the calculation time varies between a few seconds and 750s. In the latter case, diagonalisation is performed on L-spaces for 0≤L≤8, the dimensions of these spaces being between 154 and 616.

SU(3) Clebsch–Gordan coefficients at large Nc

It is argued that several papers where SU(3) Clebsch–Gordan coefficients were calculated in order to describe properties of hadronic systems are, up to a phase convention, particular cases of analytic formulae derived by Hecht in 1965 in the context of nuclear physics. This is valid for irreducible representations with multiplicity one in the corresponding Clebsch–Gordan series. For multiplicity two, Hecht has proposed an alternative which can provide correct 1/Nc sub-leading orders in large Nc studies.

Tensor space representations of Temperley–Lieb algebra via orthogonal projections of rank r ≥ 1

Unitary representations of the Temperley–Lieb algebra TLN(Q) on the tensor space (ℂn)⊗N are considered. Two criteria are given for determining when an orthogonal projection matrix P of a rank r gives rise to such a representation. The first of them is the equality of traces of certain matrices and the second is the unitary condition for a certain partitioned matrix. Some estimates are obtained on the lower bound of Q for a given dimension n and rank r. It is also shown that if 4r &amp;gt; n2, then Q can take only a discrete set of values determined by the value of n2/r. In particular, the only allowed value of Q for n = r = 2 is Q=2. Finally, properties of the Clebsch–Gordan coefficients of the quantum Hopf algebra Uq(su2) are used in order to find all r = 1 and r = 2 unitary tensor space representations of TLN(Q) such that Q depends continuously on q and P is the projection in the tensor square of a simple Uq(su2) module on the subspace spanned by one or two joint eigenvectors of the Casimir operator C and the generator K of the Cartan subalgebra.

Breaking and restoration of rotational symmetry for irreducible tensor operators on the lattice

We study the breaking of rotational symmetry on the lattice for irreducible tensor operators and practical methods for suppressing this breaking. We illustrate the features of the general problem using an $\alpha$ cluster model for $^{8}$Be. We focus on the lowest states with non-zero angular momentum and examine the matrix elements of multipole moment operators. We show that the physical reduced matrix element is well reproduced by averaging over all possible orientations of the quantum state, and this is expressed as a sum of matrix elements weighted by the corresponding Clebsch-Gordan coefficients. For our $\alpha$ cluster model we find that the effects of rotational symmetry breaking can be largely eliminated for lattice spacings of $a\leq 1.7$ fm, and we expect similar improvement for actual lattice Monte Carlo calculations.

A general non-Abelian density matrix renormalization group algorithm with application to the C2 dimer.

We extend our previous work [S. Sharma and G. K.-L. Chan, J. Chem. Phys. 136, 124121 (2012)], which described a spin-adapted (SU(2) symmetry) density matrix renormalization group algorithm, to additionally utilize general non-Abelian point group symmetries. A key strength of the present formulation is that the requisite tensor operators are not hard-coded for each symmetry group, but are instead generated on the fly using the appropriate Clebsch-Gordan coefficients. This allows our single implementation to easily enable (or disable) any non-Abelian point group symmetry (including SU(2) spin symmetry). We use our implementation to compute the ground state potential energy curve of the C2 dimer in the cc-pVQZ basis set (with a frozen-core), corresponding to a Hilbert space dimension of 10(12) many-body states. While our calculated energy lies within the 0.3 mEh error bound of previous initiator full configuration interaction quantum Monte Carlo and correlation energy extrapolation by intrinsic scaling calculations, our estimated residual error is only 0.01 mEh, much more accurate than these previous estimates. Due to the additional efficiency afforded by the algorithm, the excitation energies (Te) of eight lowest lying excited states: a(3)Πu, b(3)Σg (-), A(1)Πu, c(3)Σu (+), B(1)Δg, B(') (1)Σg (+), d(3)Πg, and C(1)Πg are calculated, which agree with experimentally derived values to better than 0.06 eV. In addition, we also compute the potential energy curves of twelve states: the three lowest levels for each of the irreducible representations (1)Σg (+), (1)Σu (+), (1)Σg (-), and (1)Σu (-), to an estimated accuracy of 0.1 mEh of the exact result in this basis.

SU(3) group structure of strange flavor hadrons

We provide the isoscalar factors of the SU(3) Clebsch-Gordan series 8⊗35 which are extensions of the previous works of de Swart, McNamee and Chilton and play practical roles in current ongoing strange flavor hadron physics research. To this end, we pedagogically study the SU(3) Lie algebra, its spin symmetries, and its eigenvalues for irreducible representations. We also evaluate the values of the Wigner D functions related to the isoscalar factors; these functions are immediately applicable to strange flavor hadron phenomenology. Exploiting these SU(3) group properties associated with the spin symmetries, we investigate the decuplet-to-octet transition magnetic moments and the baryon octet and decuplet magnetic moments in the flavor symmetric limit to construct the Coleman-Glashow-type sum rules.

Radial matrix elements and the angular momentum technique

For hydrogen wave functions, it is shown that radial matrix elements of type 〈nl|r k |nl′〉 can be expressed in terms of the Clebsch-Gordan coefficients, which characterize the relation between angular momenta l and l′, but in a nonphysical region of their arguments.

Self-dual continuous series of representations for U q s l 2 $$ {\mathcal{U}}_q\left(sl(2)\right) $$ and U q o s p 1 | 2 $$ {\mathcal{U}}_q\left(osp\left(1\Big|2\right)\right) $$

We determine the Clebsch-Gordan and Racah-Wigner coefficients for continuous series of representations of the quantum deformed algebras U_q(sl(2)) and U_q(osp(1|2)). While our results for the former algebra reproduce formulas by Ponsot and Teschner, the expressions for the orthosymplectic algebra are new. Up to some normalization factors, the associated Racah-Wigner coefficients are shown to agree with the fusing matrix in the Neveu-Schwarz sector of N=1 supersymmetric Liouville field theory.