維克定理

維克定理（英語：Wick's theorem）由吉安·卡羅·威克提出，在量子場論中廣泛用於將產生及湮滅算符的連乘積轉化為該連乘積的正規序與相應的收縮之和^[1]，在格林函數方法（英語：Green's function (many-body theory)）和費曼圖的相關問題中有重要應用。

例如，高斯自由場的維克定理說，若h是純量場、${\displaystyle D(x,y)=\langle h(x)h(y)\rangle }$是傳播子，則

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

算符的收縮的定義

兩個算符 ${\displaystyle {\hat {A}}}$ 和 ${\displaystyle {\hat {B}}}$ 的收縮定義為：

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

其中 ${\displaystyle {\mathopen {:}}{\hat {O}}{\mathclose {:}}}$ 表示算符${\displaystyle {\hat {O}}}$的正規序。算符的收縮也可以用一條在${\displaystyle {\hat {A}}}$，${\displaystyle {\hat {B}}}$上方且連接它們的線來表示，例如${\displaystyle {\overset {\sqcap }{{\hat {A}}{\hat {B}}}}}$。

下面具體檢視 ${\displaystyle {\hat {A}}}$ and ${\displaystyle {\hat {B}}}$ 分別是產生算符和湮滅算符的四種情形。 ${\displaystyle N}$ 粒子體系的產生和湮滅算符分別用 ${\displaystyle {\hat {a}}_{i}^{\dagger }}$ 和 ${\displaystyle {\hat {a}}_{i}}$ 來表示。它們滿足對易（玻色子）或反對易（費米子）關係 ${\displaystyle [{\hat {a}}_{i},{\hat {a}}_{j}^{\dagger }]_{\pm }=\delta _{ij}}$，其中 ${\displaystyle \delta _{ij}}$ 是克羅內克函數。

於是有，

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

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

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

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

其中 ${\displaystyle i,j=1,\ldots ,N}$.

由於定義正則序時已經加入了必要的正負號，所以上述關係式對玻色子和費米子都成立。由上面可見，任意兩個算符的收縮不再是算符，而是一個數。

例子

任何產生和湮滅算符的連乘積都可以用該連乘積的正則序和有限對算符的收縮表示出來。這是維克定理的基礎。在具體敘述維克定理之前，先來看幾個例子。

令 ${\displaystyle {\hat {a}}_{i}}$ and ${\displaystyle {\hat {a}}_{i}^{\dagger }}$ 為玻色子的產生和湮滅算符，它們滿足下列對易關係：

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

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

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

其中 ${\displaystyle i,j=1,\ldots ,N}$, ${\displaystyle \left[{\hat {A}},{\hat {B}}\right]\equiv {\hat {A}}{\hat {B}}-{\hat {B}}{\hat {A}}}$ 是對易子， ${\displaystyle \delta _{ij}}$ 是克羅內克函數。

根據這些對易關係，就可以把任意產生和湮滅算符的連乘積表示用其正規序與有限對算符的收縮表達出來。

例1

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

對比上式的最左邊和最右邊可見， ${\displaystyle {\hat {a}}_{i}\,{\hat {a}}_{j}^{\dagger }}$ 的順序並未發生改變，只是換了一種表達方式。

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

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

${\displaystyle ={\hat {a}}_{j}^{\dagger }\,{\hat {a}}_{i}\,{\hat {a}}_{l}^{\dagger }\,{\hat {a}}_{k}+\delta _{kl}{\hat {a}}_{j}^{\dagger }\,{\hat {a}}_{i}+\delta _{ij}{\hat {a}}_{l}^{\dagger }{\hat {a}}_{k}+\delta _{ij}\delta _{kl}}$

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

${\displaystyle ={\hat {a}}_{j}^{\dagger }{\hat {a}}_{l}^{\dagger }\,{\hat {a}}_{i}{\hat {a}}_{k}+\delta _{il}{\hat {a}}_{j}^{\dagger }\,{\hat {a}}_{k}+\delta _{kl}{\hat {a}}_{j}^{\dagger }\,{\hat {a}}_{i}+\delta _{ij}{\hat {a}}_{l}^{\dagger }{\hat {a}}_{k}+\delta _{ij}\delta _{kl}}$

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

最後一行用到了不同數目的 ${\displaystyle ^{\bullet }}$ 記號來表示不同的收縮。由最後一個例子可見，從基本對易關係來將場算符表示成正規乘積與收縮之和一般來說並不是一件容易的事，而維克定理就是用來解決這個問題的。

定理的表述

一組產生和湮滅算符的乘積 ${\displaystyle {\hat {A}}{\hat {B}}{\hat {C}}{\hat {D}}{\hat {E}}{\hat {F}}\ldots }$ 可以用正規乘積和收縮表示為：

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

換句話來說，一組產生和湮滅算符的乘積等於它們的正規乘積，加上考慮所有可能的單個收縮之後的正規乘積之和，再加上考慮所有可能的兩個收縮之後的正規乘積之和，等等。

在實際應用中，往往通過將算符交換順序而將屬於同一個收縮的兩個算符寫在一起。在交換順序時需要注意的是，每次交換兩個費米子算符的前後順序時，需要引入一個負號。例如：

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

維克定理的應用

維克定理在計算場算符的真空態期望值的時候很有用。因為所有正規乘積的真空態期望值為零，而任意兩個算符的收縮根據上面的定義是一個很容易計算的數值，故任意產生算符與湮滅算符的連乘積的真空態期望值可以很容易計算出來，例如，上面最後一個費米子的例子，式子右邊取真空態期望值後，根據正規乘積的性質，前面五項都是零，而後兩項則可用克羅內克函數計算出來（分別為 -1 和 0），故式子左邊的算符的真空態期望值為 -1。