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I am trying to complete A. Zee's book QFT in a Nutshell and at page 15 he mentions Wick's theorem and Wick contractions. (apologies for the huge page-snip).

Why does he mean by connecting the '$2n$ points' and how does the diagram of the $6x$'s prove wick's theorem?

Also why is the final result for just $a^{-3}$ times the number of distinct Wick contractions? How does one derive this?

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Essentially, what the Wick theorem tells you is that the moments of a multivariate gaussian distribution are determinate by the second moments; for instance, for a $3D$ gaussian in $(x,y,z)$ space, the quantity $$ \langle xyzx \rangle $$ can be calculated in terms of $\langle xy\rangle $, $\langle xz \rangle $, $\langle xx\rangle $ and $\langle yz\rangle $. You can indicate these quantities with the contraction symbol that you see in the book; for instance, $\langle xy\rangle = x \, y$ with a bar connecting x and y that I don't know how to put here in MathJax. With the Wick's theorem you find $$ \langle xyzx\rangle = \langle xy\rangle \langle zx\rangle +\langle xz\rangle \langle yx\rangle +\langle yz\rangle \langle xx \rangle . $$

In his example, he calculates $\langle x^6 \rangle $; he firs writes it as $\langle xxxxxx \rangle $; the only moment we need to know is $\langle x^2 \rangle $, because using the Wick theorem we can express $\langle x^6 \rangle$ as a sum of terms like $$ \langle xx\rangle \langle xx\rangle \langle xx\rangle = (\langle x^2 \rangle)^3 = a^{-3};$$ how many of them? Counting all the possible contractions like he did we get fifteen of such terms, and so $$ \langle x^6 \rangle = 15 \, a^{-3}. $$

(Note that in all of these calculations we're talking of gaussians with first moments all equal to zero)

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