So, I am trying to derive the Feynman rules for Yukawa theory (following the section in Peskin). Specifically, for the process 2 fermions $\rightarrow$ 2 fermions. To second order, I then have that the amplitude is given by

$$(-i\lambda)^2\int d^4 x \int d^4y\ _0 \langle p',k'|T \left\{\bar{\psi}(x)\psi(x)\phi(x) \bar{\psi}(y)\psi(y)\phi(y)\right\}| p,k \rangle_0 $$

where $|p,k\rangle_0$ denotes a momentum eigenstate of the free dirac hamiltonian.

Now, I am perfectly comfortable with the statement that, from Wick's theorem, that this is the sum over all possible contractions of the fields (and in this case, the states as well).

So now pretend I'm Feynman (only more charming). Is my logic just simply "wow, I'd really like to replace these contraction symbols with something a bit more visual", and then proceed to list off exactly how I'm translating between my new symbols and the contractions?

I guess I'm just having trouble coming up with a "proof" going from the statement "the sum over all contractions" to the statement "draw a bunch of pictures, following these rules and you'll get back the same thing". Do I just try a bunch of different rules until I see a pattern?

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    $\begingroup$ I think maybe an LSZ route and path integral approach to constructing Feynman diagrams would elucidate it better for you, as opposed to from Wick's theorem. $\endgroup$ – JamalS May 22 '18 at 22:50
  • $\begingroup$ Okay, I'll take a look into it. The issue is that my QFT class is using Peskin, so I'd like to keep at least the derivation within the context of that. $\endgroup$ – InertialObserver May 22 '18 at 22:51
  • $\begingroup$ Yes, just do one or two examples until you see the pattern. It’s totally possible to write a formal proof that the rules you get are the right ones, but it’s rather messy. $\endgroup$ – knzhou May 22 '18 at 23:22
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    $\begingroup$ Factors for vertices are always easy, you read them from the Hamiltonian. External lines are harder, you get these from either seeing the pattern or, as said already, formally by LSZ. Everything else is basically always the same. $\endgroup$ – knzhou May 22 '18 at 23:22
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    $\begingroup$ Some of these comments look like they should be posted as answers instead. $\endgroup$ – David Z May 23 '18 at 6:31

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