Correlation functions vs. Scattering Amplitude

[wam_mi]

Hi there,

I am learning the basics of QFT at the moment. Could someone explain to me, in the case for any number of scalar fields, the difference between correlation functions and scattering amplitudes please?

Correlation functions <0|T(\phi_{1}...\phi{n})|0>

Does one always write the mathematical expression in terms of coordinate space? What does it actually measure? Once we have calculated it, what is its significance?

Scattering Amplitude?

I read that one always need to impose momentum conservation at each vertex. But does this still hold when we have undetermined loop momentum between these vertices? What does the scattering amplitude actually measure? How is it related to the correlation function?

Many thanks!

 

[xepma]

Correlation functions constitute the fundamental objects of interest in a quantum field theory. They are, in some sense, the ultimate object you would like to determine in a given quantum field theory. The reason is that these object constitute everything physical about a quantum field theory. Any outcome of an experiment is ultimately linked to the correlation functions of the corresponding QFT. Knowing these objects means you have solved the theory, and you will notice that all the stuff you read about in a QFT book (symmetries, perturbation theory, feynman diagrams, renormalization) is ultimately linked to these functions.

This also applies to scattering amplitudes. These amplitudes are associated to the scattering processes of, for instance, particles colliding in a particle accelerator, or electrons bouncing of a plate of gold. To be more precise, in a scattering expriment you are dealing with (i) an incoming state, followed by a (ii) scattering event and ending with some (iii) outgoing state. The scattering amplitude tells you what the amplitude ('chance') is of measuring some final state given some incoming state.

A scattering amplitude is therefore an example of a correlation function. You have something physical which you can measure (the outgoing state, e.g. the momenta of the outgoing particles or the type of particles in your final state) and you have knowledge of the incoming state as well. So repeating the experiment will ultimately reveal the possible outcomes of the experiment, which, hopefully, are predicted by your QFT.

In the widest possible sense you can even argue that all experiments are of the type "incoming state from the past" followed by some "scattering event" and ending with some "outgoing state". This is why scattering amplitudes and correlation functions are frequently interchanged in high-energy theory (where you are only dealing with scattering experiments anyway)

As for your final question: correlation functions cannot be calculated using exact methods. This is where perturbation theory comes in. Perturbation theory is actually very similar to a Taylor expansion: the correlation function is some expansion in terms of an expansion parameter: the coupling constant. Feynman diagrams are a neat way of keeping track of the different terms which arise in this expansion.

 

[UndeadCat]

When you have loop momenta, you still have to impose momentum conservation everywhere- you just have to integrate the undetermined momenta, since momentum conservation won't give you the values, only relationships with the external momenta. If you're not using it already, www.damtp.cam.ac.uk/user/tong/qft.html are probably the best introductory notes, if you're cool with Fourier transforms and lagrangian dynamics; www.damtp.cam.ac.uk/user/ho/Notes.pdf continue with info about how to do loop momenta integrals etc.

 

[The_Duck]

I am learning the basics of QFT at the moment. Could someone explain to me, in the case for any number of scalar fields, the difference between correlation functions and scattering amplitudes please?

The relation between correlation functions and scattering amplitudes is given by the LSZ reduction formula, which I think should be discussed in any QFT textbook. When we are computing scattering amplitudes, we can think of correlation functions in the following way: Some of the field operators in the correlation function hit the right-hand vacuum state turn it into the "initial state" of the scattering process. Meanwhile the other field operators hit the left-hand vacuum state and turn it into the "final state" of the scattering process. So the correlation function represents the inner product of the initial and final states, which is (in the Heisenberg picture) the amplitude to go from the initial state to the final state--that is, the scattering amplitude. The LSZ reduction formula just tells you how to use field operators acting on the vacuum set up physical initial and final states consisting of particles with definite momenta. So it tells you what correlation functions to look at to extract the scattering amplitudes you want.

Correlation functions <0|T(\phi_{1}...\phi{n})|0>

Does one always write the mathematical expression in terms of coordinate space? What does it actually measure? Once we have calculated it, what is its significance?

You can also have momentum space correlation functions, which are just the Fourier transforms of position space correlation functions.

Roughly speaking, <0|phi(a)phi(b)phi(x)phi(y)|0> represents something like the amplitude that a particle at position x and a particle at positions y will propagate to positions a and b. So correlation functions are closely related to scattering amplitudes. As I said above, the LSZ reduction formula formalizes this.

I read that one always need to impose momentum conservation at each vertex. But does this still hold when we have undetermined loop momentum between these vertices?

Yes. Any QFT textbook should have a derivation of the Feynman rules. If you work through that you'll see that momentum conservation is required even when there are undetermined loop momenta.

What does the scattering amplitude actually measure?

The scattering amplitude's modulus-squared is a probability (or at least a probability density) that a certain scattering process will occur, given the initial state you are considering. You might review the simpler case of scattering off a potential in simple non-relativistic quantum mechanics. There we also talk about scattering amplitudes, and the term has the same meaning there.