The SU6 Clebsch-Gordan coefficients for the reduction of (35) ⊗ (35) to (35), and (56) ⊗ (35) to (56)

A convenient definition of the basis vectors of SU₆ representations is given. The most important Clebsch-Gordan coefficients are calculated.     

Combinatorial computation of Clebsch-Gordan coefficients

The addition of angular momenta can be reduced to elementary coupling processes of spin--particles. In this way, a method is developed which allows for a non-recursive, simultaneous computation of all Clebsch-Gordan coefficients concerning the addition of two angular momenta. The relevant equations can be interpreted easily, analogously to simple probabilistic considerations. They provide an improved understanding of the addition of angular momenta as well as a practicable evaluation of Clebsch-Gordan coefficients in an easier way than within the well-known methods.     

A Spinor Approach to the SU(2) Clebsch-Gordan Coefficients *

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.     

The Clebsch-Gordan coefficients ofsl(2,R) in the parabolic basis

We realize the Lie algebra sl(2,R) in terms of second-order differential operators defined on a dense common domain of square-integrable functions on a two-chart space, where the self-adjoint extension(s) (families) lead to all (and only) self-adjoint irreducible representations of the algebra, single- as well as multi-valued over the group. This allows for a rather straight-forward evaluation of the Clebsch-Gordan coefficients of sl(2,R) in the parabolic subalgebra basis.Invited talk presented at the International Symposium Selected Topics in Quantum Field Theory and Mathematical Physics, Bechyn, Czechoslovakia, June 14–19, 1981.I would like to thank the hospitality of Prof. J. A. de Azcárraga, at the Instituto de Fí'sica Teórica, Universidad de Valencia, where this review was written.     

Clebsch-Gordan系数的对称性

利用Schur引理确定了角动量耦合的Clebsch-Gordan系数按照其定义所允许的任意程度,然后利用正交归一关系和相位知识,导出了Clebsch-Gordan系数的对称性.     

Single particle SU(3) parentage coefficients

The single particle SU(3) parentage coefficients are calculated for the case of leading SU(3) representation in the highest orbital symmetry partition, using the method suggested by Hecht. Tabulations are given for all possible cases of identical nucleons in η=3 and η=4 shells.     

SU (4) × SU (4) symmetry breaking and its implications for SU (3) × SU (3) breaking

We investigate models where the SU (4) × SU (4) symmetry breaking Hamiltonian, $\text{H}$_SB, belongs to the (15, 15) and $\text{(10,}\text{10}\text{) + (}\text{10}\text{, 10)}$ representations, and show how they are equivalent to models of SU (3) × SU (3) breaking where $\text{H}$_SB belongs to a mixture of different representations. The results for the ππ scattering lengths in the (15, 15) model are outside the experimental limits, but the $\text{(10,}\text{10}\text{) + (}\text{10}\text{, 10)}$ model yields solutions with a wide range of values for the scattering lengths within the experimental bounds.     

Clebsch-Gordan Coefficients for the Two-Parameter Deformed Quantum Algebra SU (1,1)q,s (I)

The Bargmann representations in the tensor product space of the irreducible representations for the two-parameter deformed quantum algebra SU(1,1)_q,s corresponding to the positive discrete series (a) are in trod uced, and the corresponding Bargmann expressions for the bases of irreps, the coherent state and the operators are also derived. The Clebsch-Gordan coefficients (CGC) for the two-parameter deformed quan tum algebra SU(1,1)_q,s corresponding to the positive discrete series (a) are obtained.     

Clebsch-Gordan coefficients for the Poincaré group

All Clebsch-Gordan coefficients for the Poincaré group are calculated by a simple global method.     

Clebsch-Gordan coefficients of the groups SO(p, 1)

An explicit expression for the Clebsch-Gordan coefficients for the coupling of most degenerate unitary representations of SO(p, 1) is obtained.     

The Clebsh-Gordan series for SU(4)

A method for finding the reduction of the direct products of two irreducible representations of SU(4) is formulated by tensor method. This formula is applied for some particular cases.     

A simple sum formula for Clebsch-Gordan coefficients

I derive a simple identity involving a sum of Clebsch-Gordan coefficients with zero magnetic quantum numbers.     

Transition between SU(4) and SU(2) Kondo effect

Motivated by experiments in nanoscopic systems, we study a generalized Anderson, which consist of two spin degenerate doublets hybridized to a singlet by the promotion of an electron to two conduction bands, as a function of the energy separation δ$\delta$ between both doublets. For δ=0$\delta =0$ or very large, the model is equivalent to a one-level SU(N$N$) Anderson model, with N=4$N=4$ and 2 respectively. We study the evolution of the spectral density for both doublets (ρ_1σ(ω)${\rho }_{1\sigma }(\omega )$ and ρ_2σ(ω)${\rho }_{2\sigma }(\omega )$) and their width in the Kondo limit as δ$\delta$ is varied, using the non-crossing approximation (NCA). As δ$\delta$ increases, the peak at the Fermi energy in the spectral density (Kondo peak) splits and the density of the doublet of higher energy ρ_2σ(ω)${\rho }_{2\sigma }(\omega )$ shifts above the Ferrmi energy. The Kondo temperature T_K (determined by the half-width at half maximum of the Kondo peak in density of the doublet of lower energy ρ_1σ(ω)${\rho }_{1\sigma }(\omega )$) decreases dramatically. The variation of T_K with δ$\delta$ is reproduced by a simple variational calculation.     

Operator equivalents without the use of Clebsch-Gordan coefficients

The operator equivalent O^k_q is expressed as the qth difference of O^k_o for which a two-term recurrence relation is derived. The differences contain simple products of Stirling numbers and binomial coefficients.     

Asymptotically free SU(2) × SU(2) × SU(4) model of unified interaction

An asymptotically free SU(2) × SU(2) × SU(4) model of unified interaction is constructed. The gauge coupling constants are taken equal (in the order of magnitude) to the electromagnetic one. The strong interaction and the quark confinement is provided for by a gluonic mechanism. Some physical consequence of the model are briefly discussed.     

New approach in theory of Clebsch-Gordan coefficients foru(n) andU q(u(n))

A new method for calculation of Clebsch-Gordan coefficients (CGCs) of the Lie algebrau(n) and its quantum analogU _q(u(n)) is developed. The method is based on the projection operator method in combination with the Wigner-Racah calculus for the subalgebrau(n−1) (U _q(u(n−1))). The key formulas of the method are couplings of the tensor and projection operators and also a tensor form of the projection operator ofu(n) andU _q(u(n)). It is shown that theU _q(u(n)) CGCs can be presented in terms of theU _q(u)(n−1)) q−9j-symbols. Presented at the 9th International Colloquium: “Quantum Groups and Integrable Systems”, Prague, 22–24 June 2000. Supported by Russian Foundation for Fundamental Research, grant 99-01-01163. Supported in part by the U.S. National Science Foundation under Grant PHY-9970769 and Cooperative Agreement EPS-9720652 that includes matching from the Louisiana Board of Regents Support Fund.     

SU(4) weak currents

We suggest SU_L(4) ? U(1) as the gauge symmetry of weak and electromagnetic interactions for quartets of quarks and leptons. We analyze how the (additional) SU_L(4) weak currents (besides the SU_L(2) subgroup) could affect the weak interactions of ordinary particles, the atomic parity violation, the neutral-current neutrino reactions and the decays of the τ heavy lepton and the charmed mesons. The suppression of neutral-current parity violation in atomic experiments can be naturally incorporated in this model while at the same time the success of the Weinberg-Salam model with respect to the inclusive neutral current data is kept. The model has limited freedom and therefore many definite predictions.     

Finite transformations of SU(4)

The problem of determining the representation matrices of SU(4) is investigated. A convenient set of parameters is first introduced by writing the general element of the group as a product of exponential functions of the generators, and the generators are expressed as differential operators involving these parameters. Special matrix elements of finite transformations with a SU(3) singlet as the initial state are then obtained by solving the eigenvalue equation of the quadratic Casimir operator of SU(4). The solution has the form of a product of elementary functions and threed _mm ^j functions of SU(2) and is free from summation over intermediate states. By expanding one of thed _mm ^j functions in an appropriate series a sum rule for the special matrix elements of the permutation operator 12343412 is obtained. The discussions are strictly confined to SU(4), but, some of the results given here can be extended to unitary groups of higher dimensions.