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Ref: Yang–Mills existence and mass gap

Can anyone please explain why the Yang–Mills Existence and Mass Gap problem is so important / fundamental to contemporary mathematics (and, presumably, theoretical / mathematical physics)?

If possible, please include the following in your answer / response:

(1) Why the problem is so fundamental / important - for both mathematics and theoretical / mathematical physics; (2) How the solution to the problem would impact research in the above two fields; (3) (If applicable) What progress has been made so far in resolving it; (references to papers, books, other resources would be helpful).

I'm an upper-level undergraduate in theoretical physics, and so am seeking an answer / response at such level.

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2 Answers

Dear Anonymous, gauge theories such as QCD are among the easiest theories to be formulated yet richest theories when it comes to the phenomena they cover, many of which are important in the real world.

The existence of the mass gap - the absence of arbitrarily small positive values in the mass spectrum - is a simple property of QCD that holds but that hasn't been rigorously demonstrated.

To demonstrate it and win the $1 million from the Clay Institute, she has to define the quantum field theory at a rigorous mathematical level and master much of its physics in an equally rigorous way. So it's a good, simple enough to be formulated, mathematical problem whose solution would bring mathematicians' mastery to a higher level.

At the same moment, the paper that would win the $1 million award would almost certainly not be very important for physicists. Physicists have found lots of complementary ways and insights that made them sure that the mass gap exists. Harboring doubts about the mass gap or trying to "totally" eliminate these doubts is simply not what theoretical physicists in this discipline spend most of their man-hours.

The evidence that the mass gap is real comes from renormalization group calculations of the strength of various interactions; simulations in lattice QCD; and, among other approaches, the most modern tools are based on the holographic AdS/CFT correspondence. These physical insights are arguably much more important and valuable than whatever could be included in the hypothetical math paper that proves the existence of the mass gap.

So I would summarize this answer by saying that despite the positive hype I started with, and despite the prize that has been offered, a fully mathematical proof of the mass gap is not one of the most important problems in physics - and maths. When it comes to maths, I personally view it as much less profound than e.g. the Riemann Hypothesis. When it comes to physics, I could enumerate hundreds of problems that are more important than a rigorous proof of the mass gap.

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I'd add (parahrasing Fecko) that while everyone agrees that mathematical physics and rigorous argumentation is important for physics, the opinion of how much it is important in each concrete case varies between 0 and 100 percent :) –  Marek May 11 '11 at 8:48
And so does the definition of rigor in theoretical physics. ;-) –  Luboš Motl May 11 '11 at 9:11
@LubošMotl: Just read the statement. Why does it say a quantum yang-mills theory on $\mathbb{R}^4$? Shouldn't is be the Minkowski space? –  ramanujan_dirac Dec 15 '13 at 9:21
$R^4$ could sometimes mean the Minkowski space as well but the $1 million problem is really about the Euclidean flat space, see the first quote at en.wikipedia.org/wiki/Yang%E2%80%93Mills_existence_and_mass_gap –  Luboš Motl Dec 17 '13 at 8:31
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The Yang Mill problem ask Deep Mathematical,and physical questions directly relating to Prime numbers theories,and function analysis It addresses if Field ,waves,and particles are continous or divisibly integrated ,and how and to what degree or number of relationship between functions for energy matter,and time.Mathematical to rigorously solve such a physical equation you will need the most advance calculus not only are you dealing with multiple values,but you're also representing multiple vectors of those variables infinitely.Specifically solving yang mill field gradient flow to avoid singularities of Local field anomalies like chiral or mirror symmetry subatomic anomalies.Lot of research has been done on gradient flow like Ricci flow for certain types of Manifold metric like Riemann or Kahler.These further research areas will give deeper insight into the true structure for compacting subatomic particles wave field symmetry for generally all dimensions and spaces.

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This is a mess. Can you please edit for spacing around punctuation, capitalization, and if you can grammar. –  Brandon Enright Feb 6 at 4:40
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