Connecting n-Point Functions, Feynman Diagrams & S-Matrix

[Icosahedron]

I would like to clearify the connection between n-point functions, Feynman diagrams and S-matrix.

I do not want ( yet) to understand the detailed derivation of the LSZ reduction formula, just the rough connection between the n-point function on the one side and the S- matrix on the other side.

They say that the n-point function ( of an interacting theory) is the sum of all possible Feynman diagrams. What then do single Feynman graphs correspond to? Elements of the S-matrix? If so, how?

Thank you

 

[lbrits]

LSZ only chops off external propagators, so is no big deal (although the derivation of it is quite awful).

The n-point function, after you throw LSZ at it, represents an element of the S-matrix with given number of in- and out-states. What do single Feynman diagrams correspond to, then? Nothing, really. It's a bit like asking what the "x^3" term in the taylor series of sin(x) represents.

That being said, you can group Feynman diagrams according to say, powers of the coupling constant, or some other parameter, and treat the whole series of diagrams approximately by keeping only finitely many terms. This is in fact what is normally done. But let's say you're looking at scattering of two particles at tree level. This usually corresponds to three diagrams, (s, t, u channel). Only the sum of the three has physical meaning (the amplitude for that process, truncated to lowest order in the coupling).

 

[reilly]

Feynman diagrams are simply a 1 to 1 graphical representation of terms in covariant perturbation theory, usually for computing the S-matrix.(They are also used in non-rel theory)

ibrits' "Nothing " is not quite correct.

Why not go back to the original? -- Feynman's early papers make the subject quite clear -- also his Dover books, and many, many books on QFT. A n point function means (generally) the vacuum expectation value of n local fields -- given that any S-matrix element may be formulated as a vacuum expectation value, the set of n-point functions includes many S-matrix elements.

Regards,
Reilly Atkinson

 

[lbrits]

I disagree. Diagrams at a certain order have meaning and correspond to something physical, but many diagrams contribute *at a given order in the perturbation series*, and singling out one of them and asking "what does this diagram represent" is not always a good question. For example, IR divergences, gauge invariance, anomaly cancellation.

 

[Icosahedron]

For concretness, in $$\phi^{4}$$ theory, we calculating $$\phi\phi$$$$\rightarrow\phi\phi$$ scattering, right?

What do the matrix elements $$\langle {p_{1}p_{2}|iT|k_{1}k_{s}\rangle$$ stand for?

It is one process, why (many different) matrix elements?

 

[lbrits]

The matrix elements represent different possible momenta for the two incident particles.

 

[Icosahedron]

The matrix elements represent different possible momenta for the two incident particles.

Right. But how do I determine all those different momenta from the 4-point function?

 

[lbrits]

Well, it's the other way around. Anyway, you normally specify the incoming momenta, and integrate over the outgoing momenta. The momentum dependence of the 4pt function shows up in it's Fourier transform as well as from the LSZ formalism

 

[Icosahedron]

Thanks Ibris so far. Not yet totally clear. I will now reread chapter 4 of P&S, then come back.

Further comments in this thread are very welcome!