An Explicit and Simple Relationship Between Two Model Spaces

The paper contains the derivation of a general set of recurrence formulas for the calculus of the SU(3) Clebsch-Gordan coefficients. The first six sections are introductory, presenting the notations and placing SU(3) in the framework of the general group theory. The following eight sections are devoted to the detailed treatment of the carrier spaces of the irreducible representations and their direct product.

With the couplings between the eight gluons constrained by the structure constants of the su(3) algebra in QCD, one would expect that there should exist a special basis (or set of bases) for the algebra wherein, unlike in a Cartan-Weyl basis, {\em all} gluons interact identically (cyclically) with each other, explicitly on an equal footing. We report here particular such bases, which we have found in a computer search, and we indicate associated $3 \times 3$ representations. ...

Basis states and generator matrix elements are given for the generic representation $(a,b)$ of $G_2$ in an $SU(3)$ basis.

Alternative canonical methods for defining canonical SO(3)-coupled bases for SU(3) irreps are considered and compared. It is shown that a basis that diagonalizes a particular linear combination of SO(3) invariants in the SU(3) universal enveloping algebra gives basis states that have good $K$ quantum numbers in the asymptotic rotor-model limit.

We write a generating function for all spherical functions on the product of several copies of SU(2).

A generating function for the Wigner's $D$-matrix elements of $SU(3)$ is derived. From this an explicit expression for the individual matrix elements is obtained in a closed form.

We develop a simple computational tool for $SU(3)$ analogous to Bargmann's calculus for $SU(2)$. Crucial new inputs are, (i) explicit representation of the Gelfand-Zetlin basis in terms of polynomials in four variables and positive or negative integral powers of a fifth variable (ii) an auxiliary Gaussian measure with respect to which the Gelfand-Zetlin states are orthogonal but not normalized (iii) simple generating functions for generating all basis states and also all inva...

A presentation of the problem of calculating the vector coupling coefficients for $SU3 \supset SU2 \otimes U1$ is made, in the spirit of traditional treatments of SU2 coupling. The coefficients are defined as the overlap matrix element between product states and a coupled state with good SU3 quantum numbers. A technique for resolution of the outer degeneracy problem, based upon actions of the infinitesimal generators of SU3 is developed, which automatically produces vector co...

Using the generating function of SU(n) we find the conjugate state of SU(n) basis and we find in terms of Gel'fand basis of SU(3(n-1)) the representation of the invariants of the Kronecker products of SU(n). We find a formula for the number of the elementary invariants of SU(n). We apply our method to the coupling of SU(3) and we find a new expression of the isoscalar of Wigner symbols ($\lambda 10,\lambda 2 \mu 2; \lambda 3 \mu 3$).

A specific algebraic coupling model involving multiple quantization axes is presented in which previously indistinguishable SU(2) symmetry groups become distinguishable when coupled into a SU(3) group structure. The model reveals new intrinsic angular momentum (or isospin) eigenvectors whose structural symmetries are detailed, some of which are not available for groups having only one quantization axis available for configurations. Additionally, an intrinsic cyclic ordering o...