# 维克定理

${\displaystyle \langle h(x_{1})\ldots h(x_{2k})\rangle =\sum _{\text{对 }}D(x_{i_{1}}x_{i_{2}})\ldots D(x_{i_{2k-1}}x_{i_{2k}})}$

## 算符的收缩的定义

${\displaystyle {\hat {A}}^{\bullet }\,{\hat {B}}^{\bullet }\equiv {\hat {A}}\,{\hat {B}}\,-{\mathopen {:}}{\hat {A}}\,{\hat {B}}{\mathclose {:}}}$

${\displaystyle {\hat {a}}_{i}^{\bullet }\,{\hat {a}}_{j}^{\bullet }={\hat {a}}_{i}\,{\hat {a}}_{j}\,-{\mathopen {:}}\,{\hat {a}}_{i}\,{\hat {a}}_{j}\,{\mathclose {:}}\,=0}$
${\displaystyle {\hat {a}}_{i}^{\dagger \bullet }\,{\hat {a}}_{j}^{\dagger \bullet }={\hat {a}}_{i}^{\dagger }\,{\hat {a}}_{j}^{\dagger }\,-\,{\mathopen {:}}{\hat {a}}_{i}^{\dagger }\,{\hat {a}}_{j}^{\dagger }\,{\mathclose {:}}\,=0}$
${\displaystyle {\hat {a}}_{i}^{\dagger \bullet }\,{\hat {a}}_{j}^{\bullet }={\hat {a}}_{i}^{\dagger }\,{\hat {a}}_{j}\,-{\mathopen {:}}\,{\hat {a}}_{i}^{\dagger }\,{\hat {a}}_{j}\,{\mathclose {:}}\,=0}$
${\displaystyle {\hat {a}}_{i}^{\bullet }\,{\hat {a}}_{j}^{\dagger \bullet }={\hat {a}}_{i}\,{\hat {a}}_{j}^{\dagger }\,-{\mathopen {:}}\,{\hat {a}}_{i}\,{\hat {a}}_{j}^{\dagger }\,{\mathclose {:}}\,=\delta _{ij}}$

## 例子

${\displaystyle {\hat {a}}_{i}}$  and ${\displaystyle {\hat {a}}_{i}^{\dagger }}$  为玻色子的产生和湮灭算符，它们满足下列对易关系：

${\displaystyle \left[{\hat {a}}_{i}^{\dagger },{\hat {a}}_{j}^{\dagger }\right]=0}$
${\displaystyle \left[{\hat {a}}_{i},{\hat {a}}_{j}\right]=0}$
${\displaystyle \left[{\hat {a}}_{i},{\hat {a}}_{j}^{\dagger }\right]=\delta _{ij}}$

### 例1

${\displaystyle {\hat {a}}_{i}\,{\hat {a}}_{j}^{\dagger }={\hat {a}}_{j}^{\dagger }\,{\hat {a}}_{i}+\delta _{ij}={\hat {a}}_{j}^{\dagger }\,{\hat {a}}_{i}+{\hat {a}}_{i}^{\bullet }\,{\hat {a}}_{j}^{\dagger \bullet }=\,{\mathopen {:}}\,{\hat {a}}_{i}\,{\hat {a}}_{j}^{\dagger }\,{\mathclose {:}}+{\hat {a}}_{i}^{\bullet }\,{\hat {a}}_{j}^{\dagger \bullet }}$

### 例2

${\displaystyle {\hat {a}}_{i}\,{\hat {a}}_{j}^{\dagger }\,{\hat {a}}_{k}=({\hat {a}}_{j}^{\dagger }\,{\hat {a}}_{i}+\delta _{ij}){\hat {a}}_{k}={\hat {a}}_{j}^{\dagger }\,{\hat {a}}_{i}\,{\hat {a}}_{k}+\delta _{ij}{\hat {a}}_{k}={\hat {a}}_{j}^{\dagger }\,{\hat {a}}_{i}\,{\hat {a}}_{k}+{\hat {a}}_{i}^{\bullet }\,{\hat {a}}_{j}^{\dagger \bullet }{\hat {a}}_{k}=\,{\mathopen {:}}\,{\hat {a}}_{i}\,{\hat {a}}_{j}^{\dagger }{\hat {a}}_{k}\,{\mathclose {:}}+{\mathopen {:}}\,{\hat {a}}_{i}^{\bullet }\,{\hat {a}}_{j}^{\dagger \bullet }\,{\hat {a}}_{k}{\mathclose {:}}}$

### 例3

${\displaystyle {\hat {a}}_{i}\,{\hat {a}}_{j}^{\dagger }\,{\hat {a}}_{k}\,{\hat {a}}_{l}^{\dagger }=({\hat {a}}_{j}^{\dagger }\,{\hat {a}}_{i}+\delta _{ij})({\hat {a}}_{l}^{\dagger }\,{\hat {a}}_{k}+\delta _{kl})}$
${\displaystyle ={\hat {a}}_{j}^{\dagger }\,{\hat {a}}_{i}\,{\hat {a}}_{l}^{\dagger }\,{\hat {a}}_{k}+\delta _{kl}{\hat {a}}_{j}^{\dagger }\,{\hat {a}}_{i}+\delta _{ij}{\hat {a}}_{l}^{\dagger }{\hat {a}}_{k}+\delta _{ij}\delta _{kl}}$
${\displaystyle ={\hat {a}}_{j}^{\dagger }({\hat {a}}_{l}^{\dagger }\,{\hat {a}}_{i}+\delta _{il}){\hat {a}}_{k}+\delta _{kl}{\hat {a}}_{j}^{\dagger }\,{\hat {a}}_{i}+\delta _{ij}{\hat {a}}_{l}^{\dagger }{\hat {a}}_{k}+\delta _{ij}\delta _{kl}}$
${\displaystyle ={\hat {a}}_{j}^{\dagger }{\hat {a}}_{l}^{\dagger }\,{\hat {a}}_{i}{\hat {a}}_{k}+\delta _{il}{\hat {a}}_{j}^{\dagger }\,{\hat {a}}_{k}+\delta _{kl}{\hat {a}}_{j}^{\dagger }\,{\hat {a}}_{i}+\delta _{ij}{\hat {a}}_{l}^{\dagger }{\hat {a}}_{k}+\delta _{ij}\delta _{kl}}$
${\displaystyle =\,{\mathopen {:}}{\hat {a}}_{i}\,{\hat {a}}_{j}^{\dagger }\,{\hat {a}}_{k}\,{\hat {a}}_{l}^{\dagger }\,{\mathclose {:}}+{\mathopen {:}}\,{\hat {a}}_{i}^{\bullet }\,{\hat {a}}_{j}^{\dagger }\,{\hat {a}}_{k}\,{\hat {a}}_{l}^{\dagger \bullet }\,{\mathclose {:}}+{\mathopen {:}}\,{\hat {a}}_{i}\,{\hat {a}}_{j}^{\dagger }\,{\hat {a}}_{k}^{\bullet }\,{\hat {a}}_{l}^{\dagger \bullet }\,{\mathclose {:}}+{\mathopen {:}}\,{\hat {a}}_{i}^{\bullet }\,{\hat {a}}_{j}^{\dagger \bullet }\,{\hat {a}}_{k}\,{\hat {a}}_{l}^{\dagger }\,{\mathclose {:}}+\,{\mathopen {:}}{\hat {a}}_{i}^{\bullet }\,{\hat {a}}_{j}^{\dagger \bullet }\,{\hat {a}}_{k}^{\bullet \bullet }\,{\hat {a}}_{l}^{\dagger \bullet \bullet }{\mathclose {:}}}$

## 定理的表述

{\displaystyle {\begin{aligned}{\hat {A}}{\hat {B}}{\hat {C}}{\hat {D}}{\hat {E}}{\hat {F}}\ldots &={\mathopen {:}}{\hat {A}}{\hat {B}}{\hat {C}}{\hat {D}}{\hat {E}}{\hat {F}}\ldots {\mathclose {:}}\\&\quad +\sum _{\text{singles}}{\mathopen {:}}{\hat {A}}^{\bullet }{\hat {B}}^{\bullet }{\hat {C}}{\hat {D}}{\hat {E}}{\hat {F}}\ldots {\mathclose {:}}\\&\quad +\sum _{\text{doubles}}{\mathopen {:}}{\hat {A}}^{\bullet }{\hat {B}}^{\bullet \bullet }{\hat {C}}^{\bullet \bullet }{\hat {D}}^{\bullet }{\hat {E}}{\hat {F}}\ldots {\mathclose {:}}\\&\quad +\ldots \end{aligned}}}

${\displaystyle {\begin{array}{ll}{\hat {f}}_{1}\,{\hat {f}}_{2}\,{\hat {f}}_{1}^{\dagger }\,{\hat {f}}_{2}^{\dagger }\,&=\,{\mathcal {N}}({\hat {f}}_{1}\,{\hat {f}}_{2}\,{\hat {f}}_{1}^{\dagger }\,{\hat {f}}_{2}^{\dagger })\\&-\,{\overline {{\hat {f}}_{1}{\hat {f}}_{1}^{\dagger }}}{\mathcal {N}}({\hat {f}}_{2}\,{\hat {f}}_{2}^{\dagger })+\,{\overline {{\hat {f}}_{1}{\hat {f}}_{2}^{\dagger }}}{\mathcal {N}}({\hat {f}}_{2}\,{\hat {f}}_{1}^{\dagger })+\,{\overline {{\hat {f}}_{2}\,{\hat {f}}_{1}^{\dagger }}}{\mathcal {N}}({\hat {f}}_{1}\,{\hat {f}}_{2}^{\dagger })-{\overline {{\hat {f}}_{2}\,{\hat {f}}_{2}^{\dagger }}}{\mathcal {N}}({\hat {f}}_{1}\,{\hat {f}}_{1}^{\dagger })\\&-{\overline {{\hat {f}}_{1}\,{\hat {f}}_{1}^{\dagger }}}\,\,{\overline {{\hat {f}}_{2}\,{\hat {f}}_{2}^{\dagger }}}+{\overline {{\hat {f}}_{1}\,{\hat {f}}_{2}^{\dagger }}}\,\,{\overline {{\hat {f}}_{2}\,{\hat {f}}_{1}^{\dagger }}}\end{array}}}$

## 参考文献

1. ^ 尹道乐，尹澜. 2. 凝聚态量子理论. ISBN 9787301161609.