For two operators and we define their contraction to be
where denotes the normal order of an operator . Alternatively, contractions can be denoted by a line joining and , like .
We shall look in detail at four special cases where and are equal to creation and annihilation operators. For particles we'll denote the creation operators by and the annihilation operators by .
They satisfy the commutation relations for bosonic operators , or the anti-commutation relations for fermionic operators where denotes the Kronecker delta.
We then have
where .
These relationships hold true for bosonic operators or fermionic operators because of the way normal ordering is defined.
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 and are bosonic operators satisfying the commutation relations:
where , denotes the commutator, and is the Kronecker delta.
We can use these relations, and the above definition of contraction, to express products of and in other ways.
Example 1
Note that we have not changed but merely re-expressed it in another form as
Example 2
Example 3
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In the last line we have used different numbers of symbols to denote different contractions. By repeatedly applying the commutation relations it takes a lot of work to express in the form of a sum of normally ordered products. It is an even lengthier calculation for more complicated products.
Luckily Wick's theorem provides a shortcut.