I was trying to find the scattering amplitude using the LSZ formula for a trivial process i.e. applying it to the two-point correlation function, but I kept getting 0 as the answer.

I'm not sure exactly if this is correct or not, and, if it is, how to interpret why $\langle p_f | p_i \rangle = 0$ physically.


1 Answer 1


The $S$-matrix is often split into a trivial identity part, plus a non-trivial $T$-matrix called the "transfer matrix", for this exact reason.


Furthermore, when calculating scattering amplitudes we can factor out a general momentum-conserving delta-function from $T$:

$$T=(2\pi )^4 \delta^4\left(\sum p^\mu_f - \sum p^\mu_i\right) \mathcal{M}$$

The whole framework of calculating scattering amplitudes using Feynman rules and LSZ is for calculating $i\mathcal M$. One must keep in mind how $\mathcal M$ and $S$ are related when calculating cross sections or decay rates, which directly involve $S$.

For more you may check out chapter 5 of Schwartz.

  • $\begingroup$ I have now done lots of reading on this. I have a followup question related to when $S = 1 + iT$ holds physics.stackexchange.com/questions/691054/… $\endgroup$
    – Alex Gower
    Commented Jan 26, 2022 at 23:10
  • $\begingroup$ @AlexGower I just commented a link to a PSE which may answer that. $\endgroup$ Commented Jan 27, 2022 at 21:52

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