Wick's theorem is a method of reducing high-order derivatives to a combinatorics problem. It is named after Italian physicist Gian Carlo Wick. It is used extensively in quantum field theory to reduce arbitrary products of creation and annihilation operators to sums of products of pairs of these operators. This allows for the use of Green's function methods, and consequently the use of Feynman diagrams in the field under study. A more general idea in probability theory is Isserlis' theorem. In perturbative quantum field theory, Wick's theorem is used to quickly rewrite each time ordered summand in the Dyson series as a sum of normal ordered terms. In the limit of asymptotically free ingoing and outgoing states, these terms correspond to Feynman diagrams.
Definition of contraction For two operators A ^ {\displaystyle {\hat {A}}} and B ^ {\displaystyle {\hat {B}}} we define their contraction to be
A ^ ∙ B ^ ∙ ≡ A ^ B ^ − : A ^ B ^ : {\displaystyle {\hat {A}}^{\bullet }\,{\hat {B}}^{\bullet }\equiv {\hat {A}}\,{\hat {B}}\,-{\mathopen {:}}{\hat {A}}\,{\hat {B}}{\mathclose {:}}}
where : O ^ : {\displaystyle {\mathopen {:}}{\hat {O}}{\mathclose {:}}} denotes the normal order of an operator O ^ {\displaystyle {\hat {O}}} . Alternatively, contractions can be denoted by a line joining A ^ {\displaystyle {\hat {A}}} and B ^ {\displaystyle {\hat {B}}} , like A ^ B ^ ⊓ {\displaystyle {\overset {\sqcap }{{\hat {A}}{\hat {B}}}}} . We shall look in detail at four special cases where A ^ {\displaystyle {\hat {A}}} and B ^ {\displaystyle {\hat {B}}} are equal to creation and annihilation operators. For N {\displaystyle N} bosonic or fermionic modes we'll denote the creation operators by a ^ i † {\displaystyle {\hat {a}}_{i}^{\dagger }} and the annihilation operators by a ^ i {\displaystyle {\hat {a}}_{i}} ( i = 1 , 2 , 3 , … , N ) {\displaystyle (i=1,2,3,\ldots ,N)} . They satisfy the commutation relations for bosonic operators [ a ^ i , a ^ j † ] = δ i j 1 ^ {\displaystyle [{\hat {a}}_{i},{\hat {a}}_{j}^{\dagger }]=\delta _{ij}{\hat {\mathbf {1} }}} , or the anti-commutation relations for fermionic operators { a ^ i , a ^ j † } = δ i j 1 ^ {\displaystyle \{{\hat {a}}_{i},{\hat {a}}_{j}^{\dagger }\}=\delta _{ij}{\hat {\mathbf {1} }}} where δ i j {\displaystyle \delta _{ij}} denotes the Kronecker delta and 1 ^ {\displaystyle {\hat {\mathbf {1} }}} denotes the identity operator. We then have
a ^ i ∙ a ^ j ∙ = a ^ i a ^ j − : a ^ i a ^ j : = 0 {\displaystyle {\hat {a}}_{i}^{\bullet }\,{\hat {a}}_{j}^{\bullet }={\hat {a}}_{i}\,{\hat {a}}_{j}\,-{\mathopen {:}}\,{\hat {a}}_{i}\,{\hat {a}}_{j}\,{\mathclose {:}}\,=0}
a ^ i † ∙ a ^ j † ∙ = a ^ i † a ^ j † − : a ^ i † a ^ j † : = 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}
a ^ i † ∙ a ^ j ∙ = a ^ i † a ^ j − : a ^ i † a ^ j : = 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}
a ^ i ∙ a ^ j † ∙ = a ^ i a ^ j † − : a ^ i a ^ j † : = δ i j 1 ^ {\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}{\hat {\mathbf {1} }}}
where i , j = 1 , … , N {\displaystyle i,j=1,\ldots ,N} . These relationships hold true for bosonic operators or fermionic operators because of the way normal ordering is defined.
Examples We can use contractions and normal ordering to express any product of creation and annihilation operators as a sum of normal ordered terms. This is the basis of Wick's theorem. Before stating the theorem fully we shall look at some examples. Suppose a ^ i {\displaystyle {\hat {a}}_{i}} and a ^ i † {\displaystyle {\hat {a}}_{i}^{\dagger }} are bosonic operators satisfying the commutation relations:
[ a ^ i † , a ^ j † ] = 0 {\displaystyle \left[{\hat {a}}_{i}^{\dagger },{\hat {a}}_{j}^{\dagger }\right]=0}
[ a ^ i , a ^ j ] = 0 {\displaystyle \left[{\hat {a}}_{i},{\hat {a}}_{j}\right]=0}
[ a ^ i , a ^ j † ] = δ i j 1 ^ {\displaystyle \left[{\hat {a}}_{i},{\hat {a}}_{j}^{\dagger }\right]=\delta _{ij}{\hat {\mathbf {1} }}}
where i , j = 1 , … , N {\displaystyle i,j=1,\ldots ,N} , [ A ^ , B ^ ] ≡ A ^ B ^ − B ^ A ^ {\displaystyle \left[{\hat {A}},{\hat {B}}\right]\equiv {\hat {A}}{\hat {B}}-{\hat {B}}{\hat {A}}} denotes the commutator, and δ i j {\displaystyle \delta _{ij}} is the Kronecker delta. We can use these relations, and the above definition of contraction, to express products of a ^ i {\displaystyle {\hat {a}}_{i}} and a ^ i † {\displaystyle {\hat {a}}_{i}^{\dagger }} in other ways.
Example 1
a ^ i a ^ j † = a ^ j † a ^ i + δ i j = a ^ j † a ^ i + a ^ i ∙ a ^ j † ∙ = : a ^ i a ^ j † : + a ^ i ∙ a ^ j † ∙ {\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 }}
Note that we have not changed a ^ i a ^ j † {\displaystyle {\hat {a}}_{i}\,{\hat {a}}_{j}^{\dagger }} but merely re-expressed it in another form as : a ^ i a ^ j † : + a ^ i ∙ a ^ j † ∙ {\displaystyle \,{\mathopen {:}}\,{\hat {a}}_{i}\,{\hat {a}}_{j}^{\dagger }\,{\mathclose {:}}+{\hat {a}}_{i}^{\bullet }\,{\hat {a}}_{j}^{\dagger \bullet }}
Example 2
a ^ i a ^ j † a ^ k = ( a ^ j † a ^ i + δ i j ) a ^ k = a ^ j † a ^ i a ^ k + δ i j a ^ k = a ^ j † a ^ i a ^ k + a ^ i ∙ a ^ j † ∙ a ^ k = : a ^ i a ^ j † a ^ k : + : a ^ i ∙ a ^ j † ∙ a ^ k : {\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 {:}}+{\mathclose {:}}\,{\hat {a}}_{i}^{\bullet }\,{\hat {a}}_{j}^{\dagger \bullet }\,{\hat {a}}_{k}{\mathclose {:}}}
Example 3
a ^ i a ^ j † a ^ k a ^ l † = ( a ^ j † a ^ i + δ i j ) ( a ^ l † a ^ k + δ k l ) {\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})}
= a ^ j † a ^ i a ^ l † a ^ k + δ k l a ^ j † a ^ i + δ i j a ^ l † a ^ k + δ i j δ k l {\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}}
= a ^ j † ( a ^ l † a ^ i + δ i l ) a ^ k + δ k l a ^ j † a ^ i + δ i j a ^ l † a ^ k + δ i j δ k l {\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}}
= a ^ j † a ^ l † a ^ i a ^ k + δ i l a ^ j † a ^ k + δ k l a ^ j † a ^ i + δ i j a ^ l † a ^ k + δ i j δ k l {\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}}
= : a ^ i a ^ j † a ^ k a ^ l † : + : a ^ i ∙ a ^ j † a ^ k a ^ l † ∙ :