盖尔曼矩阵

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默里·盖尔曼

盖尔曼矩阵\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}

这八个\lambda_i矩阵是厄米的,满足对易关系:

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

其中,

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

上面出现的g_i是按照“归一化”条件

Tr(g_i g_i) = 1/2 \,

重新定义的盖尔曼矩阵,是物理中常用的归一化形式。

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[编辑]

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

参考资料[编辑]

  • Lie algebras in particle physics, by Howard Georgi (ISBN 0-7382-0233-9)
  • George Arfken and Hans Weber. Mathematical Methods for Physicists. Harcourt/Academic Press, 2000.

外部链接[编辑]