Clebsch-Gordan Research Articles

New Clebsch–Gordan‐type integrals involving threefold products of complete elliptic integrals

Multiple elliptic integrals related to the generalized Clebsch–Gordan (CG) integral are of importance in many areas in physics and special functions theory. Zhou has introduced and applied Legendre function‐based techniques to prove symbolic evaluations for integrals of CG form involving twofold and threefold products of complete elliptic integral expressions, and this includes Zhou's remarkable proof of an open problem due to Wan. The foregoing considerations motivate the results introduced in this article, in which we prove closed‐form evaluations for new CG‐type integrals that involve threefold products of the complete elliptic integrals and . Our methods are based on the use of fractional derivative operators, via a variant of a technique we had previously referred to as semi‐integration by parts.

Improved recursive computation of clebsch-Gordan coefficients

Fast, accurate, and stable computation of the Clebsch-Gordan (C-G) coefficients is always desirable, for example, in light scattering simulations, the translation of the multipole fields, quantum physics and chemistry. Current recursive methods for computing the C-G coefficients are often unstable for large quantum numbers due to numerical overflow or underflow. In this paper, we present an improved method, called the sign-exponent recurrence, for the recursive computation of C-G coefficients. The result shows that the proposed method can significantly improve the stability of the computation without losing its efficiency, producing accurate values for the C-G coefficients even with very large quantum numbers.

A Spinor Approach to the SU(2) Clebsch-Gordan Coefficients**Supported by the National Natural Science Foundation of China under Grant No. 11475016, and by the Scientific Research Foundation for Returned Scholars, Ministry of Education of China

We review the irreducible representation of an angular momentum vector operator constructed in terms of spinor algebra. We generalize the idea of spinor approach to study the coupling of the eigenstates of two independent angular momentum vector operators. Utilizing the spinor algebra, we are able to develop a simple way for calculating the SU(2) Clebsch-Gordan (CG) coefficients. The explicit expression for the SU(2) CG coefficients is worked out, and some simple physical examples are presented to illustrate the spinor approach.

Schwinger's model of angular momentum with GUP

We study the generalized uncertainty principle (GUP) modified simple harmonic oscillator (SHO) in the operator formalism by considering the appropriate form of the creation and annihilation operators . The angular-momentum algebra is then constructed using Schwinger's model of angular momentum with two independent GUP modified SHOs. With the GUP modified angular-momentum algebra, we discuss coupling of angular momentum for a two-particle composite system. Further, we calculate the Clebsch-Gordan (CG) coefficients for a two-particle system explicitly. Our results show that the CG coefficients do not receive any corrections up to quadratic GUP.

Clebsch–Gordan coefficients of discrete groups in subgroup bases

We express each Clebsch–Gordan (CG) coefficient of a discrete group as a product of a CG coefficient of its subgroup and a factor, which we call an embedding factor. With an appropriate definition, such factors are fixed up to phase ambiguities. Particularly, they are invariant under basis transformations of irreducible representations of both the group and its subgroup. We then impose on the embedding factors constraints, which relate them to their counterparts under complex conjugate and therefore restrict the phases of embedding factors. In some cases, the phase ambiguities are reduced to sign ambiguities. We describe the procedure of obtaining embedding factors and then calculate CG coefficients of the group [Formula: see text] in terms of embedding factors of its subgroups [Formula: see text] and [Formula: see text].

SUSY SU(5)×S 4 GUT flavor model for fermion masses and mixings with adjoint, large θ 13 PMNS

We propose an $S_{4}$ flavor model based on supersymmetric (SUSY) SU(5) GUT. The first and third generations of \textbf{10} dimensional representations in SU(5) are all assigned to be $1_{1}$ of $S_{4}$. The second generation of \textbf{10} is to be $1_{2}$ of $S_{4}$. Right-handed neutrinos of singlet \textbf{1} and three generations of $\bar{\textbf{5}}$ are all assigned to be $3_{1}$ of $S_{4}$. The VEVs of two sets of flavon fields are allowed a moderate hierarchy, that is $\langle\Phi^{\nu}\rangle \sim \lambda_{c}\langle\Phi^{e}\rangle$. Tri-Bimaximal (TBM) mixing can be produced at both leading order (LO) and next to next to leading order (NNLO) in neutrino sector. All the masses of up-type quarks are obtained at LO. We also get the bottom-tau unification $m_{\tau}=m_{b}$ and the popular Georgi-Jarlskog relation $m_{\mu}=3m_{s}$ as well as a new mass relation $m_{e}=\frac{8}{27}m_{d}$ in which the novel Clebsch-Gordan (CG) factor arises from the adjoint field $H_{24}$. The GUT relation leads to a sizable mixing angle $\theta^{e}_{12} \sim \theta_{c}$ and the correct quark mixing matrix $V_{CKM}$ can also be realised in the model. The resulting CKM-like mixing matrix of charged leptons modifies the vanishing $\theta^{\nu}_{13}$ in TBM mixing to a large $\theta^{PMNS}_{13}\simeq\theta_{c}/\sqrt{2}$, in excellent agreement with experimental results. A Dirac CP violation phase $\phi_{12}\simeq\pm\pi/2$ is required to make the deviation from $\theta^{\nu}_{12}$ small. We also present some phenomenological numerical results predicted by the model.

GENERALIZED HARDI INVARIANTS BY METHOD OF TENSOR CONTRACTION.

Invariants play an important role in diffusion MRI (dMRI). They represent tissue properties, such as diffusion anisotropy, and are used for registration, tissue segmentation and classification, as well as white matter integrity measures in clinical studies of debilitating brain diseases. Importantly, these features need to be rotation invariant to capture orientation independent tissue properties, as well as to enable comparisons across images that are not completely aligned. Modelling diffusion using second-order tensors, as in DTI, enable construction of diverse anatomically meaningful scalars, such as fractional anisotropy (FA), mean diffusivity (MD), and relative anisotropy (RA), which capture tissue microstructure, and are used to indicate anatomical changes [1]. Most of these scalars are derived using the tensor eigenvalues, hence, they are naturally rotation invariant as the eigenvalues remain intact under rotation. In HARDI, however, the situation is more complicated as the diffusion profile is described by higher-order tensors or orientation distribution functions (ODFs), and there is no straightforward generalization of DTI invariants to these models. The generalized FA (GFA), for example, aims to represent anisotropy in HARDI models, but has limited classification power, and is sensitive to noise being directly computed from a discrete representation of orientation distribution function (ODF) [2]. In order to take advantage of the enhanced modelling capabilities of HARDI compared to DTI, it is important to derive new invariants that capture tissue properties, and can be used as white matter biomarkers. Over the last years, researchers have created new invariants for HARDI-based models, for example, the generalized anisotropy (GA) and scaled anisotropy (SE) [3], as well as several approaches that are based on second and fourth-order tensor representations [4, 5]. A recent approach uses the Gaunt coefficients to construct invariants for ODFs and HARDI signals using the more general SH representation [2]. Our proposed technique follows a similar path of using the SH representation. We then leverage the idea of invariants constructed by tensor contraction used in computer vision for 3D pattern recognition [6]. The original idea emerged from the theory of angular momentum addition in quantum physics, and although it relies on deep and complex theoretical foundations, the formulation enables systematic construction of invariants in an elegant and simple way. This method is general, as it enables extraction of invariants from any 3D object represented as a SH series. Therefore, it can be used to construct invariants from the dMRI signal, or from any diffusion modelling object, such as ODF or FOD. In addition, it enables direct construction of invariants for any expansion order, thus, it is not limited to the common second or fourth order expansions. This method generalizes the SH descriptors used in [7] to classify autism spectrum disorder (ASD) patients and controls, and to segment brain tissue [8, 9]. It is based on constructing contravariant rank-1 tensors (vectors) using the SH and Clebsch-Gordan (CG) coefficients, and contracting them with covariant vectors to obtain rank-0 tensors (invariants). This process can be continued repeatedly to build as many invariants as desired regardless of the SH order, therefore, enables the construction of long feature vectors with strong classification capabilities. This is an advantage over the method presented in [2] in which the maximal number of invariants is bounded by the rank of the Toeplitz-like matrix. We demonstrate the strengths of our approach in both synthetic and in vivo experiments. Using simulated data we show that these invariants are robust to noise and can classify voxels based on the number of fiber compartments and their diffusivities. Using in vivo brain data, we show that they capture anatomically meaningful information, and may be used as white matter integrity measures.

Studies of electron paramagnetic resonance parameters and structure of Ni2+-vacancy center for rhombic LiCl:Ni2+ crystal

High-order perturbation formulae for the electron-paramagnetic-resonance (EPR) parameters g-factors (gx, gy and gz), and zero-field splitting (ZFS) parameters D and E of the ground state 3A2g of 3d8 ions in the crystal field (CF) with rhombic symmetry (C2v) have been derived in the strong field scheme, using the molecular orbital (MO) scheme for the first time, and applied to the investigation of Ni2+ (3d8) ions doped in LiCl crystal. In the derivation, both the contributions from all the crystal field excited states and the most important charge-transfer (CT) excited states to the EPR parameters are taken into account. The relations between the EPR parameters and the local structure parameters are established, and the defect of Ni2+-Vc center are determined quantitatively. In addition, the Clebsch-Gordan (C-G) coefficients used to construct wavefunctions in the second-kind coordinate system are calculated and given.

The IBM description of triaxial nuclei

Recently, interacting boson model (IBM) calculations, in the O(6) basis up to the seniority quantum number v max= 40 , were made possible thanks to the availability of the SO(5) Clebsch-Gordan (CG) coefficients, computed in floating-point arithmetic (T.A. Welsh, unpublished (2008)). In this paper we have made use of these CG coefficients to extend the IBM to situations where triaxial deformation is present. Such a description has been considered in the algebraic collective model (ACM) which is an algebraic version of the Bohr model. To describe triaxiality in the ACM a term proportional to cos23\( \gamma\) must be included in the Hamiltonian. We show that, in the IBM, this can be achieved by including a term quadratic in (\( \hat{{Q}}\) ⊗ \( \hat{{Q}}\) ⊗ \( \hat{{Q}}\))0, which is an IBM image of the term cos23\( \gamma\) . Quadrupole shape invariants are used to investigate the \( \beta\) and \( \gamma\) rigidity of the states obtained with such a Hamiltonian and a comparison with the rigid asymmetric rotor model is presented. The staggering of \( \gamma\) -band level energies, obtained in the present approach, is analyzed and compared to the ones predicted by the geometrical models and by the IBM. A comparison between the experimental and calculated staggerings in 190Os and 192Os is shown.

Large boson number IBM calculations and their relationship to the Bohr model

Recently, the SO(5) Clebsch-Gordan (CG) coefficients up to the seniority vmax = 40 were computed in floating point arithmetic (T.A. Welsh, unpublished (2008)); and, in exact arithmetic, as square roots of rational numbers (M.A. Caprio et al., to be published in Comput. Phys. Commun.). It is shown in this paper that extending the QQQ model calculations set up in the work by D.J. Rowe and G. Thiamova (Nucl. Phys. A 760, 59 (2005)) to N = vmax = 40 is sufficient to obtain the IBM results converged to its Bohr contraction limit. This will be done by comparing some important matrix elements in both models, by looking at the seniority decomposition of low-lying states and at the behavior of the energy and B(E2) transition strengths ratios with increasing seniority.

SO(10) group theory for the unified model building

The complete tables of Clebsch–Gordan (CG) coefficients for a wide class of SO(10) SUSY grand unified theories (GUTs) are given. Explicit expressions of states of all corresponding multiplets under standard model gauge group G321=SU(3)C×SU(2)L×U(1)Y, necessary for evaluation of the CG coefficients are presented. The SUSY SO(10) GUT model considered here includes most of the Higgs irreducible representations usually used in the literature, 10, 45, 54, 120, 126, 126¯, and 210. Mass matrices of all G321 multiplets are found for the most general superpotential. These results are indispensable for the precision calculations of the gauge couplings unification and proton decay, etc.

Programs for generating Clebsch–Gordan coefficients of SU(3) in SU(2) and SO(3) bases

Computer codes are developed to calculate Clebsch–Gordan coefficients of SU(3) in both SU(2)- and SO(3)-coupled bases. The efficiency of this code derives from the use of vector coherent state theory to evaluate the required coefficients directly without recursion relations. The approach extends to other compact semi-simple Lie groups. The codes are given in subroutine form so that users can incorporate the codes into other programs. Program summary Title of program: SU3CGVCS Catalogue identifier: ADTN Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADTN Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland Licensing provisions: Persons requesting the program must sign the standard CPC non-profit use license Computers for which the program is designed and others on which it is operable: SGI Origin 2000, HP Apollo 9000, Sun, IBM SP, Pentium Operating systems under which the program has been tested: IRIX 6.5, HP UX 10.01, SunOS, AIX, Linux Programming language used: FORTRAN 77 Memory required to execute with typical data: On the HP system, it requires about 732 KBytes. Disk space used for output: 2100+2460 bytes No. of bits in a word: 32 bit integer and 64 bit floating point numbers. No. of processors used: 1 Has the code been vectorized: No No. of bytes in distributed program, including test data, etc.: 26 309 No. of lines in distributed program, including test data, etc.: 3969 Distribution format: tar gzip file Nature of physical problem: The group SU(3) and its Lie algebra su(3) have important applications, for example, in elementary particle physics, nuclear physics, and quantum optics [1–3]. The code presented is particularly relevant for the last two fields. Clebsch–Gordan (CG) coefficients are required whenever the symmetries of many-body systems are used for the evaluation of matrix elements of tensor operators. Moreover, the construction of CG coefficients for SU(3) serves as a nontrivial prototype for larger compact semi-simple Lie algebras and even for non semi-simple Lie algebras. It is the simplest Lie algebra to have multiplicity in its outer products and a non-canonical subalgebra, i.e., SO(3). Method of solution: Vector coherent state theory is first used to construct bases for the products of two irreducible representations (irreps) [4]. The bases are SU(2)-coupled so that SU(2)-reduced CG (or isoscalar factors) can be constructed naturally. The CG coefficients in the SO(3) bases are constructed subsequently from the overlaps between the SU(2) and SO(3) bases. Restriction on the complexity of the problem: The programs are limited by computer memory and the maximum size of variable arrays. As dimension overflow conditions are possible, they are flagged and can be fixed by following the directions given as part of the error message. Typical running time: The calculation time for a single SU(3) CG coefficient is very different for SU(2) and SO(3) bases. It varies between 7.3–54.1 ns in SGI Origin 2000, 0.81–5.48 ms in HP Apollo 9000, or 0.055–0.373 ms in Intel Pentium 4 for SU(2) bases while it is between 0.027–0.255 s in Intel Pentium 4 for SO(3) bases. Unusual features of the program: Intrinsic bit functions: and, or, and shift, called iand, ior, and ishft, respectively, in FORTRAN, are used for packing and unpacking the labels for the irreps. Intrinsic logical btest is used to test the bit for the phase factor.

Clebsch-Gordan problem for the three-dimensional Lorentz group in the elliptic basis: II. Tensor products involving discrete series

Following the approach of Kerimov and Verdiyev (1998 J. Phys. A: Math. Gen. 31 3573) the Clebsch-Gordan (CG) coefficients in a SO(2) basis for tensor products involving discrete series representations of SO0(2,1) are calculated. A completeness relation for CG coefficients for all the cases under consideration is derived.

Algebraic expressions of the Clebsch-Gordon coefficients of the point group T†

A method for finding algebraic expressions of the Clebsch-Gordan (CG) coefficients of point groups is proposed and applied to the tetrahedral group. It is shown that in constructing the CG coefficients the irreducible symmetry operator (ISO) of a double group G† can be replaced by an effective ISO which is much simpler than the usual ISO. The effective ISO for the group chain T†⊃C3† is Pμ,μ̄(λ)=δμμ̄+3dμμ̄(λ)(C2z)*C2z, where d(λ)(C2z) is the matrix of C2z in the irrep λ of T†. With this effective ISO and the algebraic expression of d(λ)(C2z), the algebraic expressions are derived for the real CG coefficients of T† in the group chain T†⊃C3†. The algebraic expressions for the complex (real) CG coefficients of the group chain T†⊃D2†⊃C2† (T†⊃C2†) have also been obtained.

Clebsch-Gordan problem for three-dimensional Lorentz group in the elliptic basis: I. Tensor product of continuous series

This paper is the first of two papers devoted to the study of the Clebsch-Gordan (CG) problem for the three-dimensional Lorentz group in an elliptic (or SO(2)) basis. Here we describe the reduction of the tensor product of two unitary irreducible representations (UIRs) of the continuous series, i.e. belonging to either the principal or complementary series. The corresponding CG coefficients are defined as matrix elements of an intertwining operator between the tensor product representation and the irreducible component appearing in the decomposition. We then obtain an expression for CG coefficients in terms of a single function, namely in terms of the bilateral series with unit argument defined in the complex space of the variable . In the general case the functions are expressed in terms of two hypergeometric functions with unit argument; however, it reduces to the single function if at least one of the coupling UIRs belong to a discrete series. We derive a completeness relation for CG coefficients for all the cases under consideration.

Reduction of General Matrix Elements in the Seniority Schemeby Using a Novel Relation for CG Coefficients

Reduction relations for shell-model matrix elements of general operators in normal products ( e.g. general many-body operators) are investigated in the framework of the seniority scheme of identical particles. A novel relation for Clebsch-Gordan (CG) coefficients is exploited to remove from reduction factors all the CG coefficients that result from Wigner-Eckart theorem in the quasi-spin space. Reduction factors in the final step have very simple dependence on the number of particles.

Recursion Relations for the Clebsch-Gordan Coefficient of Quantum Group SUq(2)

Various recursion relations are exploited for the Clebsch-Gordan (CG) coefficient of the quantum group S U q (2). These are associated with four kinds of explicit forms for the CG coefficient of S U q (2).

Application of the eigenfunction method to the icosahedral group

The group table for the icosahedral group I is constructed by using the isomorphism between the group I and a subgroup of the permutation group S12. The single-valued irreducible representations and Clebsch–Gordan (CG) coefficients of I are calculated by a computer code based on the eigenfunction method. The irreducible matrix elements for all the 60 group elements are given explicitly in the form of (m/n)1/2[exp(iφ)]p [2 cos φ]q [2 cos 2φ]r, where m, n, p, q, and r are integers and φ=2π/5. The Clebsch–Gordan coefficients of I are all real under a new phase convention for time reverse states and tabulated in the form of (m/n)1/2.

Representations of the braid group obtained from quantum sl(3) enveloping algebra

The quantum Clebsch–Gordan (CG) coefficients for the coproduct 6×6 of the quantum sl(3) enveloping algebra are computed. Based on the representation 6, the representation of the braid group and the corresponding link polynomial are obtained. The link polynomials based on the representations of the quantum sl(3) enveloping algebra with a one-row Young tableau are discussed.

An Alternative Description of the Quantum GroupSUq(2) and the q-Analog Racah-Wigner Algebra

A set of generators of the quantum group S U q (2) is presented which yields the q -analog Racah-Wigner (RW) algebra. While the standard specification of S U q (2) requires separate treatment for different components of a tensor, the present approach gives unified treatment of tensor algebra. Clebsch-Gordan (CG) and Racah coefficients are extensively used. This unified treatment is crucial to formulate the q -analog irreducible tensor operator and Wigner-Eckart theorem. Witten's specification of S U q (2), exploited in rational conformal field theories (RCFT), is realized in the present approach.