Understanding mass gap in Yang-Mills theory

[Heraclitus]

Here and there some rumors come out about this relevant problem. I have read the following article in Wikipedia

http://en.wikipedia.org/wiki/Yang–Mills_existence_and_mass_gap

and the talk page

http://en.wikipedia.org/wiki/Talk:Yang–Mills_existence_and_mass_gap

but people cited there do not seem to be seriously considered as solvers for this question. Mathematics is also quite clear, as understood from the formulation of the problem given by Jaffe and Witten. On the same ground, lattice computations seem to give a sound indication of the existence of the mass gap.

So, why no fuss about this question? A recent proposed solution for P vs NP problem has been promptly assessed by the community of mathematicians. Why do not physicists do the same for solutions proposed so far?

 

[humanino]

The Yang-Mills mass gap is a mathematical problem. It will take a while before a rigorous proof is given, in fact the challenge right now mostly seems to even give a rigorous mathematical definition of the problem. I do not think it would be appropriate to make a buzz about erroneous proofs put forward every other month. Individuals with the ability to contribute to the definition and proof are already very well aware of the problem and its importance.

 

[Heraclitus]

The Yang-Mills mass gap is a mathematical problem. It will take a while before a rigorous proof is given, in fact the challenge right now mostly seems to even give a rigorous mathematical definition of the problem. I do not think it would be appropriate to make a buzz about erroneous proofs put forward every other month. Individuals with the ability to contribute to the definition and proof are already very well aware of the problem and its importance.

The point is that people claiming to have a proof have also their papers published in respectful journals and have a significant number of publications.

Last but not least, this is not just a mathematical problem as to understand how mass arises is a physics problem that has to receive a definite answer yet. Better if supported by mathematical rigor. I am just curious, as a theoretical physicist, to know how a Yang-Mills field gets confined even if I am not able to give an answer by myself.

 

[humanino]

The point is that people claiming to have a proof have also their papers published in respectful journals and have a significant number of publications.

Could you provide a reference which has been accepted by a serious journal ? There is no such publication on the wiki discussion page.

I agree with you that the mass gap is about physics. But the solution to the problem might or might not involve some kind of new physics. You asked why physicists do not make a fuss about those publications. My answer remains that the problem is a mathematical one, and that the publications I know of do not address this question directly, namely the existence of a solution, the definition of a quantum Yang-Mills gap.

 

[the_house]

I'm not an expert on the subject, but I can pretty much guarantee that any proof that has a serious chance of being correct would get plenty of attention. Confinement gets plenty of attention from physicists, and I suspect that a proof (even if it's very heavy on mathematics) would likely give much insight into the physics of confinement. Even if this isn't the case, the fact that a correct proof would garner a million dollar Millennium prize would ensure that any credible proof would receive plenty of attention.

The most likely explanation for the lack of attention payed to any claim of a proof thus far is that they are not valid. For example, the paper mentioned in the wiki talk page is almost a year and a half old and still hasn't been published, indicating that it was probably rejected from the journal it was submitted to.

 

[Heraclitus]

Could you provide a reference which has been accepted by a serious journal ? There is no such publication on the wiki discussion page.

I agree with you that the mass gap is about physics. But the solution to the problem might or might not involve some kind of new physics. You asked why physicists do not make a fuss about those publications. My answer remains that the problem is a mathematical one, and that the publications I know of do not address this question directly, namely the existence of a solution, the definition of a quantum Yang-Mills gap.

Dear Humanino,

There are a lot of reasons why some papers get overlooked that I do not want to discuss here. On the other side I do not want to become a way to publicize others papers. Anyhow, for completeness reasons I cite you some published papers that address directly this matter:

Mass gap and mass spectrum of a strongly coupled quartic scalar theory:

http://arxiv.org/abs/hep-th/0511068" appeared in Phys. Rev. D

Mass gap and mass spectrum for a strongly coupled Yang-Mills theory:

http://arxiv.org/abs/0709.2042" appeared in Phys. Lett. B

A correction to the above paper agreed between the author and Terry Tao:

http://arxiv.org/abs/0903.2357" appeared in Modern Physics Letters A

As far as I can tell, the classical mapping between the scalar field and Yang-Mills field for the gauge coupling going to infinity is proved, as also declared by Terry Tao. Indeed, this received support from lattice computations in d=2+1 from the following paper:

http://arxiv.org/abs/0912.2871" appeared in Nucl. Phys. B

What I cannot take for sure is the quantum version of this analysis and the approach of this author is not yet proved to be amenable to higher order computations. Finally, from a classical standpoint these computations are in a mathematically acceptable form being solutions to the classical equations of motion. From the side of QFT the point is to prove the existence of the generating functional in some limit. Here there is nothing to rely on.

 

[humanino]

address directly this matter

I think those papers are indeed interesting, yet they are "work in progress". None of them concludes to a solution of the millennium problem.

I cannot take for sure is the quantum version of this analysis and the approach of this author is not yet proved to be amenable to higher order computations. Finally, from a classical standpoint these computations are in a mathematically acceptable form being solutions to the classical equations of motion. From the side of QFT the point is to prove the existence of the generating functional in some limit. Here there is nothing to rely on.

It seems we agree on that. We also agree on that the problem is important and interesting. A prize by the Clay mathematical institute is appropriate, but I just do not think more "buzz" would help. It might only create more "noise".