Wick's theorem is a method of reducing high-order derivatives to a combinatorics problem.
[1] It is named after Italian physicist Gian Carlo Wick.
[2] 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.
We can use contractions and normal ordering to express any product of creation and annihilation operators as a sum of normal ordered terms.
We can use these relations, and the above definition of contraction, to express products of
By repeatedly applying the commutation relations it takes a lot of work to express
in the form of a sum of normally ordered products.
can be expressed as In other words, a string of creation and annihilation operators can be rewritten as the normal-ordered product of the string, plus the normal-ordered product after all single contractions among operator pairs, plus all double contractions, etc., plus all full contractions.
Applying the theorem to the above examples provides a much quicker method to arrive at the final expressions.
A warning: In terms on the right hand side containing multiple contractions care must be taken when the operators are fermionic.
In this case an appropriate minus sign must be introduced according to the following rule: rearrange the operators (introducing minus signs whenever the order of two fermionic operators is swapped) to ensure the contracted terms are adjacent in the string.
The contraction can then be applied (See "Rule C" in Wick's paper).
We use induction to prove the theorem for bosonic creation and annihilation operators.
base case is trivial, because there is only one possible contraction: In general, the only non-zero contractions are between an annihilation operator on the left and a creation operator on the right.
We have proved the base case and the induction step, so the theorem is true.
By introducing the appropriate minus signs, the proof can be extended to fermionic creation and annihilation operators.
The theorem applied to fields is proved in essentially the same way.
Note that the contraction of a time-ordered string of two field operators is a C-number.
In the end, we arrive at Wick's theorem: The T-product of a time-ordered free fields string can be expressed in the following manner: Applying this theorem to S-matrix elements, we discover that normal-ordered terms acting on vacuum state give a null contribution to the sum.
We conclude that m is even and only completely contracted terms remain.
This is analogous to the corresponding Isserlis' theorem in statistics for the moments of a Gaussian distribution.
Note that this discussion is in terms of the usual definition of normal ordering which is appropriate for the vacuum expectation values (VEV's) of fields.
[4]) There are any other possible definitions of normal ordering, and Wick's theorem is valid irrespective.
However Wick's theorem only simplifies computations if the definition of normal ordering used is changed to match the type of expectation value wanted.
That is we always want the expectation value of the normal ordered product to be zero.
For instance in thermal field theory a different type of expectation value, a thermal trace over the density matrix, requires a different definition of normal ordering.