I'm reading the book "Quantum Field Theory and the Standard Model" by Matthew Schwartz and I'm finding it quite hard to understand one derivation he does. It is actually short - two pages - so I find it instructive to post the pages here:(adsbygoogle = window.adsbygoogle || []).push({});

The point is that the author is doing this derivation without any of the tools that would be needed. He doesn't use the LSZ formula, nor Wick's theorem, nor perturbation theory, nor Feynman Diagrams, absolutely nothing. Also, he doesn't use spinors, nor Dirac fields in order to deal with this.

Actually he discusses this prior to introducing all these things. Now, how he does this derivation is quite confusing for me.

I mean, he first talks about a propagator that would be [itex]1/k^2[/itex]. Now I know that the Fourier transform of the classical Klein Gordon propagator actually is [itex]1/k^2[/itex], but I can't understand where this enters the discussion here, nor how this [itex]k[/itex] he associates to a propagator is the total four-momentum.

I also don't understand this discussion that leads to the formula for [itex]\mathcal{M}[/itex]. Up to this point the only thing the author has told about [itex]\mathcal{M}[/itex] is that it is related to the S-matrix by [tex]\langle f | S- \mathbf{1}|i\rangle = i(2\pi)^4 \delta^4\left(\sum p_i^\mu - \sum p_f^\mu \right) \langle f | \mathcal{M} | i\rangle[/tex] being [itex] |i\rangle, |f\rangle[/itex] respectively the initial and final states.

Only in the next chapter he derives the LSZ formula that tells how to compute [itex]S[/itex] and hence [itex]\mathcal{M}[/itex] in terms of correlation functions, and only in the chapter after that he derives Wick's theorem to finaly compute this perturbatively.

So what is really going on here? How to understand this derivation the author presents? What is the point with this [itex]1/k^2[/itex] propagator and why it relates to the center of mass energy? How this all leads to the formula for [itex]\mathcal{M}[/itex]?

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# I How to understand the derivation for this process in QFT?

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