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Clay Mathematics Institute writes about the Yang-Mills and mass gap problem on this page http://www.claymath.org/millennium/Yang-Mills_Theory/:

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.

I learned that only particles with zero rest mass can move at the speed of light, so the sentence seems like a violation of special relativity. Also I don't understand what they mean by classical waves. Classically a particle with mass is a particle. There is no wave associated with it.

Can someone explain why this statement is not a violation of special relativity and what the Clay Mathematics Institute really means?

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Yang-Mills theory fundamentally describes the dynamics of a massless spin-1 particle (among other things), called gluons. These do travel at the speed of light because they are massless. Massless particles are problematic to describe theoretically, the problem being that of "no mass gap". A well-defined quantum theory requires there to be a "mass gap" (or loosely stated, non-existence of massless particles)

What saves us from this apparent paradox is a phenomenon called Confinement. This is the statement that while gluons are massless and describe interactions in Yang-Mills theory, the do not exist independently of other particles (again, loosely stated). In other words, one will never find (or measure) an individual (free) gluon. It can however exist when there are other particles present (like the quarks), but it will never be measured then. What does exist freely is a composite object called the "glueball" that is made up entirely of gluons and while the constituents of the glueball (the gluons) are itself massless, the glueball is not (due to interaction energies). Thus, the objects that do exist freely are all massive and the theory has a mass gap and all is well!

You can think of a gluon having as a classical wave travelling at speed $c$ and a glueball as a superposition of several such classical waves, whose group velocity is less than $c$.

Confinement is known to happen and it has been shown experimentally, but there is no theoretical understanding of this phenomenon yet.

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When you are saying gluon have a classical wave, do you mean gluons can be described by a classical wave equation? –  asmaier Jun 27 '13 at 15:21
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