# Scattering matrix and correlation functions

## Main Question or Discussion Point

I've been reading Coleman's notes and the book on QFT by Ticciati. There they both place a lot of emphasis on computing the scattering matrix S. I can follow their computations (using Wick's theorem etc.) but I don't really have a good understanding of what S actually tells you. Ticciati even mentions that once you have S you've solved the theory. Why is this the case? In order to get another viewpoint I started reading the QFT book by Peskin & Schroeder. Here they start with the computation of the correlation functions. How are these n-point correlation functions related to S? Thanks in advance.

## Answers and Replies

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The actual experiments that can be conducted to investigate subatomic particles is to take two beams and smash them together. Then you observe the intensity of the scattered beams at various angles, to see how much the beam was deflected by the collision. That's what the S-matrix tells you: for two given particles with defined momenta going into the collision, it tells you the quantum mechanical amplitude for any given number of particles with given momenta coming out of it. Therefore, once you're able to compute S-matrix elements, you're able to effectively predict the outcome of any collision that can be performed. Peskin and Schroeder Chapter 1 has a good overview of these sorts of experiments, and gives you sort of a 10,000 foot view on why we're doing all of this.

The main way that you compute S-matrix elements is by making use of the LSZ reduction formula. This formula requires the n-point correlation function for the theory in question (or, more realistically, a perturbative approximation to it). That's why Peskin and Schroeder spend so much time discussing it--it's the key to the S-matrix. To see how you get the S-matrix itself once you have the correlation functions, see P&S Chapter 7.

Thanks for the reply. Is it fair to say that the correlation functions are solely computed to get at the scattering amplitudes, or do they have any other significant relevance?

It is instructive not to use all the language of qft to get scattering matrix elements like using LSZ reduction formula and all the partition function analysis using the path integral formulation.It is best to extract feynman rules from the given interaction and draw them to calculate the amplitudes.If you are using wick contraction for the interaction part sandwiched between initial and final states,then this is also very transparent to calculate the scattering matrix element because it will give you the amplitudes directly.By the way,those n point green function are used for only calculating the S matrix elements.

DarMM
Science Advisor
Gold Member
Thanks for the reply. Is it fair to say that the correlation functions are solely computed to get at the scattering amplitudes, or do they have any other significant relevance?
That is their main use. They can also be computed to learn about bound states, compute energy levels of states, or to learn more about the algebraic properties of the theory, but these computations are more rarely done.

By the way, the S-matrix does not contain all the information of the field theory, so even if
you calculated S non-perturbatively, you wouldn't have solved the theory. This is because the
S-matrix is computed from the time ordered correlation (tau-)functions and these correlation functions do not contain the complete information of the field theory, i.e. if somebody handed
you these functions you couldn't rebuild the Hilbert space and the field operator from them.