标题： 一种导出 Clebsch-Gordan 系数代数公式的$SU(3)$微积分 显示英文标题

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

摘要： 我们开发了一个简单的计算工具用于$SU(3)$，类似于 Bargmann 对$SU(2)$的微积分。 关键的新输入包括：(i) 以四变量的多项式和第五变量的正整数或负整数幂显式表示 Gelfand-Zetlin 基；(ii) 一个辅助的高斯测度，Gelfand-Zetlin 态在这种测度下是正交但未归一化的；(iii) 生成所有基态和所有不变量的简单生成函数。 作为我们技术的一个实例，首次得到了 Clebsch-Gordan 系数的代数公式。 这仅涉及高斯积分。 因此，$SU(3)$在计算上变得与$SU(2)$一样易于处理。 显示英文摘要

摘要： 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.