Normal products, contractions and Wicks theorem

To make further progress, we must understand how to handle products of creation and annihilation operators of the type arising in eq. (3.91). We shall introduce Wick s theorem and also some necessary definitions and relations. 

The creation and annihilation operators satisfy the anticommutation relation 

It is this anticommutation property which ensures that the AT-particle system obeys Fermi statistics. The anticommutator allows the order of creation and annihilation operators in a product to be changed by writing it in the form 

Antisymmetry of electrons implies that annihilation operators anticommute [Xa,Xb = X Xb + XbXa 

we have to define the normal product or n-product, and contraction or pairing. The simplification of matrix elements requires that we move creation operators to the left of annihilation operators. A normal product is a reordered operator string which satisfies this requirement. A contraction or pairing of creation and/or annihilation operators is their vacuum expectation value. 

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