Wick's theorem
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 formula_1 and formula_2 we define their contraction to be where formula_4 denotes the normal order of an operator formula_5. Alternatively, contractions can be denoted by a line joining formula_1 and formula_2, like formula_8. We shall look in detail at four special cases where formula_1 and formula_2 are equal to creation and annihilation operators. For formula_11 particles we'll denote the creation operators by formula_12 and  the annihilation operators by formula_13 formula_14. They satisfy the commutation relations for bosonic operators formula_15, or the anti-commutation relations for fermionic operators formula_16 where formula_17 denotes the Kronecker delta. We then have where formula_22. 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 formula_13 and formula_12  are bosonic operators satisfying the commutation relations: where formula_22, formula_29 denotes the commutator, and formula_17 is the Kronecker delta. We can use these relations, and the above definition of contraction, to express products of formula_13 and formula_12 in other ways. Example 1. Note that we have not changed formula_34 but merely re-expressed it in another form as formula_35 Example 3. In the last line we have used different numbers of formula_42 symbols to denote different contractions. By repeatedly applying the commutation relations it takes a lot of work, as you can see, to express formula_43 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. Statement of the theorem. A product of creation and annihilation operators formula_44 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). Example: If we have two fermions (formula_46) with creation and annihilation operators formula_47 and formula_48 (formula_49) then Note that the term with contractions of the two creation  operators and of the two annihilation operators is not included because their contractions vanish. Proof of Wick's theorem. We use induction to prove the theorem for bosonic creation and annihilation operators. The formula_46 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. Suppose that Wick's theorem is true for formula_53 operators formula_54, and consider the effect of adding an "N"th operator formula_1 to the left of formula_54 to form formula_57. By Wick's theorem  applied to formula_53 operators, we have: formula_1 is either a creation operator or an annihilation operator. If formula_1 is a creation operator, all above products, such as formula_62, are already normal ordered and require no further manipulation. Because formula_1 is to the left of all annihilation operators in formula_64, any contraction involving it will be zero. Thus, we can add all contractions involving formula_1 to the sums without changing their value. Therefore, if formula_1 is a creation operator, Wick's theorem holds for formula_64. Now, suppose that formula_68 is an annihilation operator. To move formula_68 from the left-hand side  to the right-hand side of all the products, we repeatedly swap formula_1 with the operator immediately right of it (call it formula_71), each time applying formula_72 to account for noncommutativity. Once we do this, all terms will be normal ordered. All terms added to the sums by pushing formula_1 through the products correspond to additional contractions involving formula_1. Therefore, if formula_1 is an annihilation operator, Wick's theorem holds for formula_64. 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. Wick's theorem applied to fields. The correlation function that appears in quantum field theory can be expressed by a contraction on the field operators: where the operator formula_78 are the amount that do not annihilate the vacuum state formula_79. Which means that formula_80. This means that formula_81 is a contraction over formula_82. 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. where "p" is the number of interaction fields (or, equivalently, the number of interacting particles) and "n" is the development order (or the number of vertices of interaction). For example, if formula_87 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. (Wick's theorem provides as a way of expressing VEV's of "n" fields in terms of VEV's of two fields.) 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.