A Calculus for $SU(3)$ Leading to an Algebraic Formula for Clebsch-Gordan Coefficients

Abstract

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 invariants. As an illustration of our techniques, an algebraic formula for the Clebsch-Gordan coefficients is obtained for the first time. This involves only Gaussian integrations. Thus $SU(3)$ is made as accessible for computations as $SU(2)$ is.

RevTex 3.0, 104 pages, one figure. To Appear in J.M.P

Cited by in corpus (12)

• Fuzzy CP2
• SU(N) Irreducible Schwinger Bosons
• Prepotential formulation of SU(3) lattice gauge theory
• Coherent States For SU(3)
• Irreducible SU(3) Schwinger Bosons
• SU(N) Coherent States
• Invariants, Projection Operators and $SU(N)\times SU(N)$ Irreducible Schwinger Bosons
• A complementary group technique for the resolution of the outer multiplicity problem of SU(n): (II) A recoupling approach to the solution of SU(3)\supset U(2) reduced Wigner coefficients
• A complementary group technique for a resolution of the outer multiplicity problem of SU(n): (I) Littlewood rule and a complementary group of SU(n)
• Addition of SU(3) generators and its Singlet Hilbert space
• New Discrete States in Two-Dimensional Supergravity
• A modern Fortran library for SU(3) coupling and recoupling coefficients