• Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.


New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback


(propose a free ad)

Site Statistics

204 submissions , 162 unreviewed
5,026 questions , 2,180 unanswered
5,344 answers , 22,683 comments
1,470 users with positive rep
815 active unimported users
More ...

  Scattering amplitude, link between quantum mechanics and QFT

+ 3 like - 0 dislike

In quantum mechanics, we can define the scattering amplitude $f_k(\theta)$ for two particles as the magnitude of an outgoing spherical wave. More precisely, the asymptotic behaviour (when $r\rightarrow\infty$) of a wave function of two scattering particles, interacting with some short range potential, is given by

$\psi(r)=e^{ikz} + \frac{f_k(\theta)}{r}e^{ikr}$

where the incoming wave is the plane wave $e^{ikz}$ (the coordinates are the relative coordinates between the two particles). The full Hamiltonian is given by


The low energy limit can be obtained by expanding the scattering amplitude in partial waves and only include the lowest partial wave.

However, we can also compute this in effective field theory. In the low energy limit, the effective lagrangian is

$L=\psi^\dagger\left(i\frac{\partial}{\partial t}+\frac{1}{2}\nabla^2\right)\psi-\frac{g_2}{4}(\psi^\dagger \psi)^2$

We can then define the four point greens function as $\langle0|T\psi\psi\psi^\dagger\psi^\dagger|0\rangle$. We can then define the scattering amplitude A as, and I quote (see later for reference) "It is obtained by subtracting the disconnected terms that have the factored form $\langle0|T\psi\psi^\dagger|0\rangle\langle0|T\psi\psi^\dagger|0\rangle$, Fourier transforming in all coordinates, factoring out an overall energy-momentum conserving delta function, and also factoring out propagators associated with each of the four external legs". For two particles, the amplitude A only depends on the total energy E. The claim is then that we have


My question is, although I understand that it is reasonable that there is a relation between these two quantities, I have no idea how to prove this and how to get the numerical factors right etc. So basically, what is the exact link between doing scattering computations in QM vs QFT? How can one show that the observables we are looking at is the same quantity?

The paper I am following is http://arxiv.org/abs/cond-mat/0410417 . The Effective field theory part I am referring to starts at page 135 , especially the relation (295). The above quote of the definition of A is given on page 139. The definition of $f_k$ is given on page 10-11.

This post imported from StackExchange Physics at 2015-05-31 13:08 (UTC), posted by SE-user user2133437
asked May 26, 2015 in Theoretical Physics by user2133437 (55 points) [ no revision ]

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
To avoid this verification in future, please log in or register.

user contributions licensed under cc by-sa 3.0 with attribution required

Your rights