Wick's theorem

Wick's theorem is a method of reducing high-order derivatives to a combinatorics problem. It is named after 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.

Definition of contraction

For two operators $\hat{A}$ and $\hat{B}$ we define their contraction to be

where $\mathopen{:} \hat{O} \mathclose{:}$ denotes the normal order of an operator $\hat{O}$ .

Alternatively, contractions can be denoted by a line joining $\hat{A}$ and $\hat{B}$ .

We shall look in detail at four special cases where $\hat{A}$ and $\hat{B}$ are equal to creation and annihilation operators. For $N$ particles we'll denote the creation operators by $\hat{a}_i^\dagger$ and the annihilation operators by $\hat{a}_i$ ( $i=1,2,3\ldots,N$ ). They satisfy the usual commutation relations $[\hat{a}_i,\hat{a}_j^\dagger]=\delta_{ij}$ , where $\delta_{ij}$ denotes the Kronecker delta.