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

TL;DR

This paper introduces a new computational calculus for $SU(3)$, enabling algebraic calculation of Clebsch-Gordan coefficients using polynomial bases, generating functions, and Gaussian integrations, making $SU(3)$ more accessible for computations.

Contribution

It develops an $SU(3)$ calculus with explicit polynomial bases, generating functions, and Gaussian measures, leading to the first algebraic formula for Clebsch-Gordan coefficients.

Findings

Derived an explicit Gelfand-Zetlin basis in polynomial form.

Established generating functions for basis states and invariants.

Obtained an algebraic formula for $SU(3)$ 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.