# 盖尔曼矩阵

## 特殊表象

$\lambda_i$（i=1到8）表示如下：[1]:283-288

 $\lambda_1 = \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}$ $\lambda_2 = \begin{pmatrix} 0 & -i & 0 \\ i & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}$ $\lambda_3 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 0 \end{pmatrix}$ $\lambda_4 = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{pmatrix}$ $\lambda_5 = \begin{pmatrix} 0 & 0 & -i \\ 0 & 0 & 0 \\ i & 0 & 0 \end{pmatrix}$ $\lambda_6 = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix}$ $\lambda_7 = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & -i \\ 0 & i & 0 \end{pmatrix}$ $\lambda_8 = \frac{1}{\sqrt{3}} \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -2 \end{pmatrix}$

$[g_i, g_j] = if^{ijk} g_k \,$

$g_i = \frac{\lambda_i}{2} \,$

$Tr(g_i g_i) = 1/2 \,$

$f^{ijk}$关于三个指标i,j,k，是全反对称的。它们的非零分量为

$f^{123} = 1 \ , \quad f^{147} = f^{165} = f^{246} = f^{257} = f^{345} = f^{376} = \frac{1}{2} \ , \quad f^{458} = f^{678} = \frac{\sqrt{3}}{2} \ .$

## 参考文献

1. ^ Griffiths, David J., Introduction to Elementary Particles 2nd revised, WILEY-VCH, 2008, ISBN 978-3-527-40601-2

### 延伸閱讀

• Howard Georgi，Lie algebras in particle physicsISBN 0-7382-0233-9
• George Arfken，Hans Weber，Mathematical Methods for Physicists. Harcourt/Academic Press, 2000. ISBN 0123846544
• J. J. J. Kokkedee，The quark model，Frontiers in physics，ISBN 0805356118