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Generally speaking the average guy marginally acquainted with quantum field theory or advanced combinatorics describes Wick's theorem as some sort of correspondence between higher order differential calculus and combinatorics. In fact more words can be added to drive the point. I want to understand something here. Can someone tell me what Wicks theorem is in the following fashion:

  1. Meaning of differentiation in ordinary language.
  2. New and suitable language for differentiation (say as used in QFT).
  3. Higher order derivatives in this language and why it is better than traditional language.
  4. Define a few basics of combinatorics as relevant to the problem.
  5. Make the connection with combinatorics.
  6. Make connection with quantum field theory.

If you are really sure this is not the right way to teach this stuff propose an alternative way, but start the description with X* so I am sure you are talking about something else. As usual correct me where necessary.

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There is a way to understand Wick's Theorem as a particular instance of a very general connection between differentiation and combinatorics. At the bottom of it is just Leibniz's product rule. Given variables $x_1,\ldots,x_d$, the basic identity is: $$ \frac{\partial}{\partial x_{i_1}}\ldots\frac{\partial}{\partial x_{i_n}}\ x_{j_1}\ldots x_{j_n}= \sum_{\sigma\in\mathfrak{S}_n} \ \delta_{i_1,j_{\sigma(1)}}\ldots \delta_{i_n,j_{\sigma(n)}}\ . $$ Suppose you have two (not necessarily symmetric) tensors $A_{i_1,\ldots,i_n}$ and $B_{j_1,\ldots,j_n}$. Then you can contract the indices in the identity with $A$ and $B$ and this generates a sum over all ways of connecting the legs of $A$ to those of $B$.

Wick's theorem for a Gaussian measure with covariance $C$ can be written as $$ \frac{1}{(2\pi)^{\frac{n}{2}}\sqrt{{\rm det}\ C}}\int_{\mathbb{R}^d} e^{-\frac{1}{2}x^TC^{-1}x}\ x_{j_1}\ldots x_{j_n} \ dx_1\ldots dx_d=\left. \exp\left(\frac{1}{2}\partial C\partial\right) x_{j_1}\ldots x_{j_n}\ \right|_{x=0}\ . $$ In this case $A$ is given by $$ A_{i_1,\ldots,i_n}=\frac{1}{2^{\frac{n}{2}} \left(\frac{n}{2}\right)!} C_{i_1,i_2}C_{i_3,i_4}\ldots C_{i_{n-1},i_n} $$ whereas $B$ is the indicator function of the given sequence of indices $j_1,\ldots,j_n$.

This kind of connection between differentiation and sums over combinatorial structures like graphs is quite old. It goes back at least to the article "On the theory of the analytical forms called trees" by Cayley. It is on page 172 of the 1857 volume of the Philosophical Magazine.

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     I am really just motivating a few links. I also hope a few comments may be added to the mix. 

                          UPDATE: See Qmechanic's links for extended details relevant to qft.

Well I was not sure if I should dare write anything here, but I am sure there are some flaws in my thinking, so someone can correct me. OK. Let's begin with some disclaimers and disclosures. I did not invent wick's theorem and applications and extensions there of. I am not an expert. All I am posting here is a collection of ideas developed by real scientists and respectable people. I have done digging and am just reporting on findings. All credit for the correct parts of this goes to the people concerned and sites and texts that I have looked at. All errors are mine, and you can send me mail about my errors. So as far as I understand Let's consider something of the form :

$\exp{\frac{1}{2}b^T A^{-1} b}$ $\rightarrow$ $\sum A^{-1}_{ij}b^ib^j$

Essentially just rewriting the thing and preparing it for the next step

via $\exp(x) = 1 +x + \frac{x^2}{2}$ then $\rightarrow$ $\frac{1}{n!} \frac{1}{2^n}(\sum A^{-1}_{ij}b^ib^j)^n$

As I have seen, if we have some differential operator $\frac{\partial}{\partial b^k}$

Then we can say $\frac{\partial}{\partial b^k}$ $\sum A^{-1}_{ij}b^ib^j$ = $A^{-1}_{ik}b^i$

I guess the idea is what happens when you operate twice. How about n-times. I will defer to the link where I got this methodology

http://www.ams.org/samplings/feature-column/fcarc-feynman4

I think the author of this grabbed the ideas from yet another page on Geometric and Algebraic structures in Mathematics

http://www.math.sunysb.edu/events/dennisfest/


can someone mutate this line of thought for bosons and fermions

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