# 盖尔曼矩阵

## 特殊表象

${\displaystyle \lambda _{i}}$ （i=1到8）表示如下：[1]:283-288

 ${\displaystyle \lambda _{1}={\begin{bmatrix}0&1&0\\1&0&0\\0&0&0\end{bmatrix}}}$ ${\displaystyle \lambda _{2}={\begin{bmatrix}0&-i&0\\i&0&0\\0&0&0\end{bmatrix}}}$ ${\displaystyle \lambda _{3}={\begin{bmatrix}1&0&0\\0&-1&0\\0&0&0\end{bmatrix}}}$ ${\displaystyle \lambda _{4}={\begin{bmatrix}0&0&1\\0&0&0\\1&0&0\end{bmatrix}}}$ ${\displaystyle \lambda _{5}={\begin{bmatrix}0&0&-i\\0&0&0\\i&0&0\end{bmatrix}}}$ ${\displaystyle \lambda _{6}={\begin{bmatrix}0&0&0\\0&0&1\\0&1&0\end{bmatrix}}}$ ${\displaystyle \lambda _{7}={\begin{bmatrix}0&0&0\\0&0&-i\\0&i&0\end{bmatrix}}}$ ${\displaystyle \lambda _{8}={\frac {1}{\sqrt {3}}}{\begin{bmatrix}1&0&0\\0&1&0\\0&0&-2\end{bmatrix}}}$

${\displaystyle [g_{i},g_{j}]=if^{ijk}g_{k}\,}$

${\displaystyle g_{i}={\frac {\lambda _{i}}{2}}\,}$

${\displaystyle Tr(g_{i}g_{i})=1/2\,}$

${\displaystyle f^{ijk}}$ 关于三个指标i,j,k，是全反对称的。它们的非零分量为

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

## 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