维克定理

维克定理（英語：Wick's theorem）由吉安·卡罗·威克提出，在量子场论中广泛用于将产生及湮滅算符的连乘积转化为该连乘积的正规序与相应的收缩之和^[1]，在格林函数方法（英语：Green's function (many-body theory)）和费曼图的相关问题中有重要应用。

例如，高斯自由场的维克定理说，若h是纯量场、${\displaystyle D(x,y)=\langle h(x)h(y)\rangle }$是传播子，则

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

算符的收缩的定义

两个算符 ${\displaystyle {\hat {A}}}$  和 ${\displaystyle {\hat {B}}}$  的收缩定义为：

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

其中 ${\displaystyle {\mathopen {:}}{\hat {O}}{\mathclose {:}}}$  表示算符${\displaystyle {\hat {O}}}$ 的正规序。算符的收缩也可以用一条在${\displaystyle {\hat {A}}}$ ，${\displaystyle {\hat {B}}}$ 上方且连接它们的线来表示，例如${\displaystyle {\overset {\sqcap }{{\hat {A}}{\hat {B}}}}}$ 。

下面具体检视 ${\displaystyle {\hat {A}}}$  and ${\displaystyle {\hat {B}}}$  分别是产生算符和湮灭算符的四种情形。 ${\displaystyle N}$  粒子体系的产生和湮灭算符分别用 ${\displaystyle {\hat {a}}_{i}^{\dagger }}$  和 ${\displaystyle {\hat {a}}_{i}}$  来表示。它们满足对易（玻色子）或反对易（费米子）关系 ${\displaystyle [{\hat {a}}_{i},{\hat {a}}_{j}^{\dagger }]_{\pm }=\delta _{ij}}$ ，其中 ${\displaystyle \delta _{ij}}$  是克罗内克函数。

于是有，

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

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

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

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

其中 ${\displaystyle i,j=1,\ldots ,N}$ .

由于定义正则序时已经加入了必要的正负号，所以上述关系式对玻色子和费米子都成立。由上面可见，任意两个算符的收缩不再是算符，而是一个数。

例子

任何产生和湮灭算符的连乘积都可以用该连乘积的正则序和有限对算符的收缩表示出来。这是维克定理的基础。在具体叙述维克定理之前，先来看几个例子。

令 ${\displaystyle {\hat {a}}_{i}}$  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：

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

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

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

其中 ${\displaystyle i,j=1,\ldots ,N}$ , ${\displaystyle \left[{\hat {A}},{\hat {B}}\right]\equiv {\hat {A}}{\hat {B}}-{\hat {B}}{\hat {A}}}$  是对易子， ${\displaystyle \delta _{ij}}$  是克罗内克函数。

根据这些对易关系，就可以把任意产生和湮灭算符的连乘积表示用其正规序与有限对算符的收缩表达出来。

例1

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

对比上式的最左边和最右边可见， ${\displaystyle {\hat {a}}_{i}\,{\hat {a}}_{j}^{\dagger }}$  的顺序并未发生改变，只是换了一种表达方式。

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

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

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

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

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

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

最后一行用到了不同数目的 ${\displaystyle ^{\bullet }}$  记号来表示不同的收缩。由最后一个例子可见，从基本对易关系来将场算符表示成正规乘积与收缩之和一般来说并不是一件容易的事，而维克定理就是用来解决这个问题的。

定理的表述

一组产生和湮灭算符的乘积 ${\displaystyle {\hat {A}}{\hat {B}}{\hat {C}}{\hat {D}}{\hat {E}}{\hat {F}}\ldots }$  可以用正规乘积和收缩表示为：

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

换句话来说，一组产生和湮灭算符的乘积等于它们的正规乘积，加上考虑所有可能的单个收缩之后的正规乘积之和，再加上考虑所有可能的两个收缩之后的正规乘积之和，等等。

在实际应用中，往往通过将算符交换顺序而将属于同一个收缩的两个算符写在一起。在交换顺序时需要注意的是，每次交换两个费米子算符的前后顺序时，需要引入一个负号。例如：

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

维克定理的应用

维克定理在计算场算符的真空态期望值的时候很有用。因为所有正规乘积的真空态期望值为零，而任意两个算符的收缩根据上面的定义是一个很容易计算的数值，故任意产生算符与湮灭算符的连乘积的真空态期望值可以很容易计算出来，例如，上面最后一个费米子的例子，式子右边取真空态期望值后，根据正规乘积的性质，前面五项都是零，而後两项则可用克罗内克函数计算出来（分别为 -1 和 0），故式子左边的算符的真空态期望值为 -1。

参考文献

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

阅读

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• M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory, Perseus Books (1995). (§4.3)
• Tony Philips. Finite-dimensional Feynman Diagrams. What's New In Math. American Mathematical Society. November 2001 [2007-10-23]. （原始内容于2014-01-08）. 
• Emilio San Fabian. Wick's theorem. 2001年2月 [2008-07-29]. （原始内容于2008-09-19）. 
• T.S. Evans, D.A. Steer, Wick's theorem at finite temperature, Nucl. Phys B 474, 481-496 (1996) arXiv:hep-ph/9601268