# 特殊酉群

（重定向自特殊么正群

${\displaystyle \operatorname {SU} (n)}$粒子物理标准模型中有广泛的应用，特别是 ${\displaystyle \operatorname {SU} (2)}$电弱相互作用${\displaystyle \operatorname {SU} (3)}$量子色动力学中。

## 性质

SU(n) 代数由 n2 个算子生成，满足交换关系（对 i, j, k, l = 1, 2, ..., n）：

${\displaystyle \left[{\hat {O}}_{ij},{\hat {O}}_{kl}\right]=\delta _{jk}{\hat {O}}_{il}-\delta _{il}{\hat {O}}_{kj}}$

${\displaystyle {\hat {N}}=\sum _{i=1}^{n}{\hat {O}}_{ii}}$

${\displaystyle \left[{\hat {N}},{\hat {O}}_{ij}\right]=0}$

## 生成元

• ${\displaystyle \operatorname {tr} (T_{a})=0,\,}$

• ${\displaystyle T_{a}=T_{a}^{\dagger }.\,}$

### 基本表示

• ${\displaystyle T_{a}T_{b}={\frac {1}{2n}}\delta _{ab}I_{n}+{\frac {1}{2}}\sum _{c=1}^{n^{2}-1}{(if_{abc}+d_{abc})T_{c}}\,}$

• ${\displaystyle \left[T_{a},T_{b}\right]_{+}={\frac {1}{n}}\delta _{ab}+\sum _{c=1}^{n^{2}-1}{d_{abc}T_{c}}\,}$
• ${\displaystyle \left[T_{a},T_{b}\right]_{-}=i\sum _{c=1}^{n^{2}-1}{f_{abc}T_{c}}\,}$

• ${\displaystyle \sum _{c,e=1}^{n^{2}-1}d_{ace}d_{bce}={\frac {n^{2}-4}{n}}\delta _{ab}\,}$

### 伴随表示

• ${\displaystyle (T_{a})_{jk}=-if_{ajk}\,}$

## SU(2)

${\displaystyle \operatorname {SU} _{2}(\mathbb {C} )}$  一个一般矩阵元素形如

${\displaystyle U={\begin{pmatrix}\alpha &-{\overline {\beta }}\\\beta &{\overline {\alpha }}\end{pmatrix}}}$

${\displaystyle \varphi (\alpha ,\beta )={\begin{pmatrix}\alpha &-{\overline {\beta }}\\\beta &{\overline {\alpha }}\end{pmatrix}}.}$

${\displaystyle U'={\begin{pmatrix}ix&-{\overline {\beta }}\\\beta &-ix\end{pmatrix}}}$

${\displaystyle u_{1}={\begin{pmatrix}0&i\\i&0\end{pmatrix}}\qquad u_{2}={\begin{pmatrix}0&-1\\1&0\end{pmatrix}}\qquad u_{3}={\begin{pmatrix}i&0\\0&-i\end{pmatrix}}}$

${\displaystyle [u_{1},u_{3}]=2u_{2},\qquad [u_{2},u_{1}]=2u_{3},\qquad [u_{3},u_{2}]=2u_{1}.}$

## SU(3)

SU(3) 的生成元 T，在定义表示中为

${\displaystyle T_{a}={\frac {\lambda _{a}}{\sqrt {2}}}.\,}$

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

• ${\displaystyle \left[T_{a},T_{b}\right]=i\sum _{c=1}^{8}{f_{abc}T_{c}}\,}$

${\displaystyle f^{123}=1\,}$
${\displaystyle f^{147}=-f^{156}=f^{246}=f^{257}=f^{345}=-f^{367}={\frac {1}{2}}\,}$
${\displaystyle f^{458}=f^{678}={\frac {\sqrt {3}}{2}}\,}$

d 的取值：

${\displaystyle d^{118}=d^{228}=d^{338}=-d^{888}={\frac {1}{\sqrt {3}}}\,}$
${\displaystyle d^{448}=d^{558}=d^{668}=d^{778}=-{\frac {1}{2{\sqrt {3}}}}\,}$
${\displaystyle d^{146}=d^{157}=-d^{247}=d^{256}=d^{344}=d^{355}=-d^{366}=-d^{377}={\frac {1}{2}}\,}$

## 李代数

${\displaystyle \mathrm {SU} (n)}$  对应的李代数记作 ${\displaystyle {\mathfrak {su}}(n)}$ 。它的标准数学表示由无迹反埃尔米特 ${\displaystyle n\times n}$  复矩阵组成，以通常交换子李括号粒子物理学家通常增加一个因子 ${\displaystyle i}$ ，从而所有矩阵成为埃尔米特的。这只不过是同一个实李代数一个不同的更方便的表示。注意 ${\displaystyle {\mathfrak {su}}(n)}$ ${\displaystyle \mathbb {R} }$  上一个李代数。

${\displaystyle i\sigma _{x}={\begin{bmatrix}0&i\\i&0\end{bmatrix}}}$
${\displaystyle i\sigma _{y}={\begin{bmatrix}0&1\\-1&0\end{bmatrix}}}$
${\displaystyle i\sigma _{z}={\begin{bmatrix}i&0\\0&-i\end{bmatrix}}}$

（这里 ${\displaystyle i}$ 虚数单位。）

${\displaystyle iI_{2}={\begin{bmatrix}i&0\\0&i\end{bmatrix}}}$

${\displaystyle \mathrm {SU} (n)}$ ${\displaystyle n-1}$ ，它的邓肯图${\displaystyle A_{n-1}}$  给出，有 ${\displaystyle n-1}$ 顶点的链。

${\displaystyle (1,-1,0,\dots ,0)}$ ,
${\displaystyle (0,1,-1,\dots ,0)}$ ,
…,
${\displaystyle (0,0,0,\dots ,1,-1)}$ .

${\displaystyle {\begin{pmatrix}2&-1&0&\dots &0\\-1&2&-1&\dots &0\\0&-1&2&\dots &0\\\vdots &\vdots &\vdots &\ddots &\vdots \\0&0&0&\dots &2\end{pmatrix}}}$ .

## 广义特殊酉群

${\displaystyle M\in SU(p,q,R)}$

${\displaystyle M^{*}AM=A\,}$
${\displaystyle \det M=1.\,}$

${\displaystyle A={\begin{bmatrix}0&0&i\\0&I_{n-2}&0\\-i&0&0\end{bmatrix}}.}$

## 重要子群

${\displaystyle SU(n)\supset SU(p)\times SU(n-p)\times U(1).}$

${\displaystyle SU(n)\supset O(n)}$
${\displaystyle SU(2n)\supset USp(2n).}$

${\displaystyle SO(2n)\supset SU(n)}$
${\displaystyle USp(2n)\supset SU(n)}$
${\displaystyle Spin(4)=SU(2)\times SU(2)}$ （参见自旋群
${\displaystyle E_{6}\supset SU(6)}$
${\displaystyle E_{7}\supset SU(8)}$
${\displaystyle G_{2}\supset SU(3)}$ （关于 E6, E7 与 G2 参见单李群）。

## 注释

1. ^ R.R. Puri, Mathematical Methods of Quantum Optics, Springer, 2001.