# Yang–Mills and Mass Gap

#### Thales

Registered Member
This thread shall become about starting to consider the many open questions pertaining to the Yang–Mills and Mass Gap. That will be the generic "purpose" of this post, rather than attempting to cause a proof of it to appear, immediately. As such, it remains an unsolved problem in the field of 'mathematical physics', and one of seven Millennium Prize Problems as defined by the Clay Mathematics Institute. Here is a link to the CMI description of the Yang-Mills existence and mass gap problem:
The laws of quantum physics stand to the world of elementary particles in the way that Newton's laws of classical mechanics stand to the macroscopic world. Almost half a century ago, Yang and Mills introduced a remarkable new framework to describe elementary particles using structures that also occur in geometry. Quantum Yang-Mills theory is now the foundation of most of elementary particle theory, and its predictions have been tested at many experimental laboratories, but its mathematical foundation is still unclear. The successful use of Yang-Mills theory to describe the strong interactions of elementary particles depends on a subtle quantum mechanical property called the "mass gap": the quantum particles have positive masses, even though the classical waves travel at the speed of light. This property has been discovered by physicists from experiment and confirmed by computer simulations, but it still has not been understood from a theoretical point of view. Progress in establishing the existence of the Yang-Mills theory and a mass gap will require the introduction of fundamental new ideas both in physics and in mathematics.

The Clay Mathematics Institute page contains a [.pdf] of the "Official Problem Description", as well as another document concerning "The Status of The Problem" by Michael Douglas. There is additionally a supplementary lecture by Lorenzo Sadun located directly under the "Related links" sub-header. And, of course, the Wikipedia page likewise provides a working statement of the problem:
The problem is phrased as follows:[1]

Yang–Mills Existence and Mass Gap. Prove that for any compact simple gauge group G, a non-trivial quantum Yang–Mills theory exists on
and has a mass gap Δ > 0. Existence includes establishing axiomatic properties at least as strong as those cited in Streater & Wightman (1964), Osterwalder & Schrader (1973) and Osterwalder & Schrader (1975).

I recently happened upon this nLab page for "mass gap" and admittedly I am left at a bit of a "loss" since I am wholly unclear as to what specific background is requisite so as to be actually able to apprehend, properly summarize, or have any real putative access to this seemingly important issue. What happened, historiographically, in that wide area of mathematical physics?

Put differently, what should I read about this sort of 'topicality' first? How should one begin to learn 'non-perturbative quantum field theory'? That said, SciForum, I am just writing right now to sound off this initial inquiry: What are your most intelligent thoughts surrounding Yang-Mills and "mass gap"? Are there any available insights into how to best approach this problem, that you would wish to discuss?

A heartfelt "thanks"- truly - for your effort,

Thales

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My only suggestion is to try elsewhere e.g. PhysicsForums.com where genuine expert specialists may help. You won't get useful traction here.

Thanks for your suggestion. And, why not?

Thanks for your suggestion. And, why not?
Lack of expertise. We once had a string theorist of sorts as mod but he left.

One million to solve it would suggest there is great difficulty to solve it.
Alex

Will any users address my row of beginner's questions, then? Is this cited 'lack of expertise' necessarily an impasse? Who ultimately 'decides' what, or indeed who, is qualified to give an answer to the Yang-Mills and "mass gap" problem? Who stands where, exactly?

Unfortunately Yang-Mills isn't something that's easy to explain in simple terms, you really need an understanding of symmetry groups like SU(2) and SU(3), gauge theories, and so on.

Without some background in the classical theories of electromagnetism and gravity and all the mathematics involved under your belt, it will probably be mostly impenetrable. I can personally recommend for instance, reading up on SU(5) theory and why it turned out to be wrong, but that's just me.

On the other hand there are some youtube videos on the subject, and as we say in my part of the world, "see how you go mate". But hey, this is an unsolved problem (not the only one!), and if the Clay Institute is offering a prize you can bet it isn't exactly a doddle, mate.

Thanks for your comment, @arfa_brane. I appreciate it.

To reiterate: I do not count myself among the experts, nor am I necessarily "itching" to solve it. I just want to better understand. To wit, I am sure any true "string theorist" would indeed run circles with a needle in-hand around this thread. I'm simply just setting off along this winding path. However, I've been somewhat familiar with the classical theories of electromagnetism & gravity. For instance, I am usually able to directly manage the Hamiltonian/Lagrangian, without undue obstruction.

So, what are the necessary "prerequisites" for this discovery? Gauge theory?

I can personally recommend for instance, reading up on SU(5) theory and why it turned out to be wrong, but that's just me.

Would you provide some "sources" that you find credible? Here is an informal blog post I dug up this morning - On History - although it does not incorporate how SU(5) turned out to be wrong, as you put it. What happened with SU(5), that you find helpful to know?

Thanks again. I actually like that "see how you go" approach.

Will any users address my row of beginner's questions, then?
Well, it is difficult to know where to start - do you know (for example) what is meant by a gauge theory? Be honest - don't just quote the wiki back to us. Do you understand what this is?

If you do, do you understand the difference between a "general" gauge theory and the Yang-Mills theory?

Is this cited 'lack of expertise' necessarily an impasse?
Lack of expertise is (to say the least) an undesirable asset in a teaching situation. In the solution to famously difficult problems, lack of expertise is somewhat like the infinite monkeys with infinite typewriters and infinite time reproducing Hamlet

\@Thales

Michael Nielsen
is a quite readable author who I know as the co-author of a fairly weighty volume on quantum information science. He's a tenured professor at some Australian university (which one I forget just now, but not important) and a recognised authority on topics including QIS; university courses in that subject would likely include the above as a reference text.

There's an online pdf, a chapter from some book, by Nielsen which looks like a reasonable introduction to Yang-Mills theory, what it is and why it's important etc. Note, I don't profess to understand much of it myself, btw.

Should at least give you some idea of what you're looking at, or how many years of study you have ahead, perhaps . . .

One more:

Gerardus t'Hooft wrote an article for Scientific American way back in 1980, titled Gauge Theories of the Forces between Fundamental Particles.
He covers, in layman's terms, what Yang-Mills fields are, first mentioned on p9.

Thanks a lot, arfa brane. I've been scanning those science papers, to get understanding. I'm now having to reconsider the true definition of "electron", since perhaps my thinking went askew, during all these years. I'm also vaguely thinking about a passage I had read involving Emmy Noether and her "invariance" theorems; Are you able to help clarify what a paragraph such as the following one (below) indicates/identifies? What is the rapport between Noether and Yang-Mills?
Noether identities need not be independent, but satisfy first-stage Noether identities, which are subject to the second-stage Noether identities and so on. Higher-stage Noether identities also are separated into the trivial and non-trivial once. A degenerate Lagrangian is called reducible if there exist non-trivial higher-stage Noether identities. Yang–Mills gauge theory and gauge gravitation theory exemplify irreducible Lagrangian field theories.

Apologies, if I've been jumping around a bit. Thanks again for your patience.

Thales said:
Are you able to help clarify what a paragraph such as the following one (below) indicates/identifies? What is the rapport between Noether and Yang-Mills?
Not really, sorry. That would be something in the domain of postgraduate study, perhaps in mathematical physics.

You need a good grasp of continuous symmetries, Euler-Lagrange equations etc. What I know about Noether's theorems I could write on a postage stamp maybe.

What are your most intelligent thoughts surrounding Yang-Mills and "mass gap"? Are there any available insights into how to best approach this problem, that you would wish to discuss?

You can see my article titled "Structure of a Particle"( uploaded at Academia.edu), if this can help to provide some insight for the solution to this problem.

Lack of expertise. We once had a string theorist of sorts as mod but he left.
Who was the resident string theorist in this forum? Must've been after my time...

Yes Alpha was a clever guy, and (away from the forum) a good friend to me.

Also BenTheMan whose PhD was in supersymmetry , also a fun guy. Went into banking I think - what a waste of talent.

You can see my article titled "Structure of a Particle"( uploaded at Academia.edu), if this can help to provide some insight for the solution to this problem.