# LSZ formula applied to two-point correlation function

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.

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.

$$S=1+iT$$

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.

• 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/… Commented Jan 26, 2022 at 23:10
• @AlexGower I just commented a link to a PSE which may answer that. Commented Jan 27, 2022 at 21:52