• 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.


PO is now at the Physics Department of Bielefeld University!

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

205 submissions , 163 unreviewed
5,054 questions , 2,207 unanswered
5,347 answers , 22,726 comments
1,470 users with positive rep
818 active unimported users
More ...

  Conflict between Lippmann–Schwinger equation and Gell-Mann and Low theorem about energy

+ 2 like - 0 dislike

Lippmann–Schwinger equation states that scattering state will have the same energy as free state, while Gell-Mann Low theorem says that they have different enery.

Lippmann–Schwinger equation says: That is, if $|f\rangle $ is the eigenstate of interacting picture's free Hamiltonian $\hat{H}_0^{(I)}$ with eigenvalue $E$, then $$|F \rangle := \hat{U}(0,-\infty)|f\rangle$$ is the eigenstale of the full Hamiltonian $\hat{H}^{(I)}(0)$ at $t=0$ with the same eigenvalue $E$, where $\hat{U}(t,t_0)$ is the evolution operator in interacting picture.

On the other hand, Gell-Mann and Low theorem states that :

Let $|f\rangle$ be an eigenstate of $H_0$ with energy $E_0$ and let the 'interacting' Hamiltonian be $H=H_0 + gV$, where $g$ is a coupling constant and $V$ the interaction term. We define a Hamiltonian $H_\epsilon=H_0 + e^{-\epsilon |t|}gV$ which effectively interpolates between $H$ and $H_0$ in the limit $\epsilon \rightarrow 0^+$ and $|t|\rightarrow\infty$. Let $U_{\epsilon I}$ denote the evolution operator in the interaction picture. The Gell-Mann and Low theorem asserts that if the limit as $\epsilon\rightarrow 0^+$ of

$|F_\epsilon \rangle = \frac{ U_{\epsilon I} (0,-\infty) |f\rangle}{\langle f | U_{\epsilon I}(0,-\infty)|f\rangle}$ exists, then $|F_\epsilon \rangle$ are eigenstates of $H$. But the energy $E$ will not be same as $E_0$ and will have a shift.

Here is a proof of my first state that energy will be same

enter image description here

enter image description here

enter image description here

enter image description here

enter image description here

enter image description here

enter image description here

enter image description here

enter image description here

This post imported from StackExchange Physics at 2014-04-16 05:28 (UCT), posted by SE-user user34669
asked Apr 16, 2014 in Theoretical Physics by user34669 (205 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).
Please complete the anti-spam verification

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

Your rights