Wicks Theorem for the Evaluation of Matrix Elements

Using the anticommutation relations of Eqs. [19], [20], and [21], an arbitrary string of annihilation and creation operators can be written as a linear combination of normal-ordered strings (most of which contain reduced num- 

In addition, a third combination in which A is a creation operator and B is an annihilation operator is also zero, since the string is again already normal-ordered  

The final combination, in which A is an annihilation operator and B is a creation operator, is not zero, however, owing to the anticommutation relations in Eq. [21]. Thus we write 

Note that we must maintain the correct sign when the operators in the braces, are reordered. 

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