Using Wick's theorem on already normal-ordered functions The book Quantum Field Theory of Many Body systems (X. G. Wen) defines Wick's theorem as follows.

I would like to employ this definition to simplify the 4-operator product $$\hat{O}\;=\;a_p^\dagger a_q^\dagger a_r a_s$$ into a sum of two-operator products. However, as this operator is already normal ordered, the terms $W$ and $:W:$ are the same in the above expression, and therefore all other terms on the RHS would equal zero. Could someone kindly provide pointers on the correct implementation of Wick's theorem in this case?
 A: The operator is already normal-ordered and does not require any further simplification. Note that it annihilates the vacuum state. On the other hand, simplifying operator $a_s a_r a_q^\dagger a_p^\dagger$ could be a useful exercise.
A: It is already normal ordered, as you say, so there is no point in applying the normal ordering to it.
Now, I haven't studied the applications of QFT to many bodies, so I don't know what purpose Wick's theorem has in this book/area, but I don't think you should be using it to re-express an ordinary operator; i.e. istead of applying the formula you attach just apply the conmutation relations of the annihilation-creation operators to normal order it, thus deriving the factors you may or may'nt have. I only use Wick's theorem to go from time ordering from normal ordering, albeit my knowledge in the subject is very limited at present.
Of course, the two approaches are equivalent, but since you asked for pointers in applying it, I feel like you would get more familiar with it by working out the expression yourself rather than simply applying a formula.
Sorry for the lengthy -and subjective- answer, have a nice day :)
