Quantcast
  • 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.

News

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

Attributions

(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,082 questions , 2,232 unanswered
5,353 answers , 22,788 comments
1,470 users with positive rep
820 active unimported users
More ...

  What does it mean that there is no mathematical proof for confinement?

+ 10 like - 0 dislike
3252 views

I see this all the time* that there still doesn't exist a mathematical proof for confinement. What does this really mean and how would a sketch of a proof look like?

What I mean by that second question is: what are the steps one needs to prove in order to "mathematically prove confinement".


*See e.g. Scherer's "Introduction to Chiral Perturbation Theory" middle of page 7

This post imported from StackExchange Physics at 2014-08-29 16:47 (UCT), posted by SE-user Love Learning
asked Jun 11, 2014 in Theoretical Physics by Love Learning (165 points) [ no revision ]

1 Answer

+ 11 like - 0 dislike

The problem

In case you were not aware of this, finding a proof for confinement is one of the Millenium Problems by the Clay Mathematics Institute. You can find the (detailed) answer to your question in the official problem description by Arthur Jaffe and Edward Witten.

In short: proving confinement is essentially equivalent to showing that a quantum Yang-Mills theory exists and is equipped with a "mass gap". The latter manifests itself in the fact that the lowest state in the spectrum of the theory cannot have an arbitrarily low energy, but can be found at some energy $\Delta>0$. Proving this means to formulate the theory in the framework of axiomatic quantum field theory and deduce systematically all of its properties.

Mass gap implies confinement

In order to understand why proving that the theory has a mass gap is equal to proving confinement, we first have to understand what confinement is. In technical language it means that all observable states of finite energy are singlets under transformations of the global colour $\text{SU}(3)$. In simple terms this means that all observable particles are colour-neutral. Since quarks and gluons themselves carry colour charge, this implies that they cannot propagate freely, but occur only in bound states, namely hadrons.

Proving that the states in the theory cannot have arbitrarily low energies, i.e. there is a mass gap, means that there are no free particles. This in turn means that there cannot be free massless gluons which would have no lower bound on their energy. Hence, a mass gap implies confinement.

Motivation

The existence of confinement, while phenomenologically well-established, is not fully understood on a purely theoretical level. Confinement is a low energy phenomenon and is as such not accessible by perturbative QCD. There exist various low energy effective theories such as chiral perturbation theory which, while giving good phenomenological descriptions of hadron physics, do not teach us much about the underlying mechanism. Lattice QCD, albeit good for certain qualitative and quantitative predictions, also does not allows us to prove something on a fundamental level. Furthermore, there is the AdS/CFT correspondence, which allows us to describe theories which are similar to QCD in many respects, but a description of QCD itself is not accessible at this point. To conclude: there are many open questions to answer before we have a full understanding of QCD.

This post imported from StackExchange Physics at 2014-08-29 16:48 (UCT), posted by SE-user Frederic Brünner
answered Jun 11, 2014 by Frederic Brünner (1,130 points) [ no revision ]
Most voted comments show all comments
Why a non-singlet bound state imply free gluons?

This post imported from StackExchange Physics at 2014-08-29 16:48 (UCT), posted by SE-user user10001
Non-singlet is equivalent to non-confining which in turn implies having free gluons. What do you mean by non-singlet bound states?

This post imported from StackExchange Physics at 2014-08-29 16:48 (UCT), posted by SE-user Frederic Brünner
I mean a bound state which is not in singlet representation of global SU(3). Can't such states exist ?

This post imported from StackExchange Physics at 2014-08-29 16:48 (UCT), posted by SE-user user10001
Not in a confining theory.

This post imported from StackExchange Physics at 2014-08-29 16:48 (UCT), posted by SE-user Frederic Brünner
Yes, but I don't understand how the existence of mass gap can imply non-existence of such states.

This post imported from StackExchange Physics at 2014-08-29 16:48 (UCT), posted by SE-user user10001
Most recent comments show all comments
Thank you! Yes, this is only true below the confinement scale. All the statements in my answer regarding confinement and mass gap are to be thought of in the context of low energy QCD.

This post imported from StackExchange Physics at 2014-08-29 16:48 (UCT), posted by SE-user Frederic Brünner
If we assume that finite energy states are global-SU(3) singlets then it would imply that there is a mass gap. However, I don't understand how the converse is true. There may be a mass gap but, in principle, there may still exist finite energy states which are not global-SU(3) singlets. No?

This post imported from StackExchange Physics at 2014-08-29 16:48 (UCT), posted by SE-user user10001

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:
p$\hbar$ysicsOve$\varnothing$flow
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
...