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Let $\phi$ be a scalar field and then I see the following expression for the square of the normal ordered version of $\phi^2(x)$.

$T(:\phi^2(x)::\phi^2(0):) = 2<0|T(\phi(x)\phi(0))|0>^2 + 4<0|T(\phi(x)\phi(0))|0>:\phi(x)\phi(0): + :\phi^2(x)\phi^2(0):$

It would be great if someone can help derive the above expression - may be from scratch - and without outsourcing to Wick's theorem - and may be help connect as to why the above is related (equal?) to the Wick's theorem?

  • Isn't the above also known as OPE (Operator Product Exapnsion)? If yes, then is there at all any difference between OPE and Wick's theorem? Is there a systematic way to derive such OPEs?

  • Can one help extend this to Fermions?

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migrated to physics.stackexchange.com by Piotr Migdal Apr 21 '12 at 13:29

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Apologies but this question seems to be asking "please teach me Wick's theorem" - and it moreover says "do it without teaching me Wick's theorem". Have you tried to study Wick's theorem? At least en.wikipedia.org/wiki/Wick%27s_theorem ? The standard pedagogical treatment answers all your questions. The Wick's theorem is the systematic way to construct such identities that you're looking for. Fermions differ by some signs only. I find it questionable whether copying/rephrasing sections from standard textbook material is a good way to use this server and people's time. –  Luboš Motl Apr 21 '12 at 11:57