• 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
5,345 answers , 22,719 comments
1,470 users with positive rep
818 active unimported users
More ...

  Proof for the Mass gap for sine-Gordon action with $g \cos(\Phi)$

+ 5 like - 0 dislike

This is the sine-Gordon action: $$ \frac{1}{4\pi} \int_{ \mathcal{M}^2} dt \; dx \; k \partial_t \Phi \partial_x \Phi - v \partial_x \Phi \partial_x \Phi + g \cos(\ell_{}^{} \cdot\Phi_{}) $$ Here $\mathcal{M}^2$ is a 1+1 dimensional spacetime, where 1D space is a $S^1$ circle of length $L$.

At $g=0$ : it is a chiral boson theory with zero mass, gapless scalar boson $\Phi$.

At large $g$ : It seems to be well-known that at large coupling $g$ of the sine-Gordon equation, the scalar boson $\Phi$ will have a mass gap.

Q1: What is the original Ref which states and proves this statement about the nonzero (or large) mass gap for large $g$?


Q2: What does the mass gap $m$ scale like in terms of other quantities (like $L$, $g$, etc)?


NOTE: I find S Coleman has discussion in

(1)"Aspects of Symmetry: Selected Erice Lectures" by Sidney Coleman

and this paper

(2)Quantum sine-Gordon equation as the massive Thirring model - Phys. Rev. D 11, 2088 by Sidney Coleman

But I am not convinced that Coleman shows it explicitly. I read these, but could someone point out explicitly and explain it, how does he(or someone else) rigorously prove this mass gap?

Here Eq.(17) of this reference does a quadratic expansion to show the mass gap $m \simeq \sqrt{\Delta^2+\#(\frac{g}{L})^2}$ with $\Delta \simeq \sqrt{ \# g k^2 v}/(\# k)$, perhaps there are even more mathematical rigorous way to prove the mass gap with a full cosine term?

This post imported from StackExchange Physics at 2014-06-04 08:00 (UCT), posted by SE-user Idear
asked Jun 4, 2014 in Theoretical Physics by wonderich (1,500 points) [ no revision ]
Via bosonization technique you can prove that the sine Gordon model is equivalent to the neutral sector of the massive thirring model. With this technique you can also calculate exactly the correlation functions and see directly that they fall exponentially with distance so that the theory has mass gap. See also this paper by Mandelstam journals.aps.org/prd/abstract/10.1103/PhysRevD.11.3026

This post imported from StackExchange Physics at 2014-06-04 08:00 (UCT), posted by SE-user TwoBs

@ TwoBs, does that correlator argument have any β dependence of cos(βΦ)? In S Coleman, there is some discussion when it is Thirring model, when it is energy unbounded from below, when it is just a free massive Fermion model.

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