Yang-Mills Theory: Unsolved Mathematics and String Theory

[kurt.physics]

Hello, I am curious to know what yang-mills theory is? What does it say?

Clay mathematics institute has, in the year 2000, issued a $1,000,000 dollar prize for who ever solves some sort of proof for it. Does anyone here work on this?

I noticed, in the documentary "the elegant universe" that there were 2 mathematical problems, one was something, and the other was the yang-mills thing. How does this tie in with string theory, does this give the proof.

 

[AlphaNumeric2]

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

Yang Mills theory is a quantum theory which does not contain quarks or any other fermions and is only about the gauge particles generated by the SU(N) gauge group. In SU(3) Yang Mills, the requirement of local gauge invariance generates the description of gluons.

 

[blechman]

Yang Mills theory is a quantum theory which does not contain quarks or any other fermions and is only about the gauge particles generated by the SU(N) gauge group.

Speaking as a high-energy theorist, this is not true. I (and my colleagues) call any theory with a nonabelian gauge symmetry a "Yang-Mills" theory. Indeed, part of the Yang-Mills lore is precisely figuring out how to couple fermions to gauge fields through covariant derivatives and field strengths. If there are no fermions, then it's a "pure" Yang-Mills theory (what Clay is interested in). If there are only gauge fields and massless Majorana fermions in the adjoint representation of the gauge group (minimally coupled), it's called a "(pure) Super-Yang-Mills" theory. But there is no requirement for "no fermions" in the definition of Yang-Mills.

 

[BenTheMan]

As a corallary to this question: what is the mass gap, and why is it a problem?

I mean, take a U(1) Yang-Mills with a fermion (QED). I know that there is a pole in the propogator, and the pole mass gives the energy of an electron at rest. There may be some (bound) states with energies slghtly less than the pole mass, but in general there is a gap between the vacuum and the one particle zero momentum sate.

So why is this a million dollar question? QED is an effective field theory anyway, and as such it doesn't really attempt to explain its parameters, right?

 

[blechman]

As a corallary to this question: what is the mass gap, and why is it a problem?

I mean, take a U(1) Yang-Mills with a fermion (QED). I know that there is a pole in the propogator, and the pole mass gives the energy of an electron at rest. There may be some (bound) states with energies slghtly less than the pole mass, but in general there is a gap between the vacuum and the one particle zero momentum sate.

So why is this a million dollar question? QED is an effective field theory anyway, and as such it doesn't really attempt to explain its parameters, right?

Consider the Hamiltonian of a pure Yang-Mills theory. Find it's eigenstates. Show that the lowest eigenstate above the vacuum has non-vanishing energy. And you win $1M!

Sounds easy, doesn't it? I mean, what's the big deal? You learn how to solve eigensystems in your undergrad QM class, possibly even before that, so surely this won't be THAT much of a big deal...

Well, it is! The difference between QED ("trivial" YM) and the more general theories is that they "confine". That means that in a PURE SU(N) Yang-Mills theory, the eigenstates of the system are not "gluons" but rather "hadrons". So what you need to do is to somehow express the Yang-Mills theory of gluons in terms of a basis of hadrons. Nobody has ANY idea how to do this. Even your QED example is not quite as trivial as it sounds: you can only define the mass perturbatively. That is, the propogator with its pole is only defined in terms of a perturbative expansion. Thus your "proof" of the mass of the electron is somewhat flawed - you've only defined it in a perturbative sense - and the Clay people want a full, nonperturbative solution.

The situation in QCD is even more complicated, since there are fermions (quarks). Perturbatively, you can show that QCD still has the property of "asymtotic freedom" that confines quarks and gluons to hadrons. But if there are too many fermions, then it will no longer hold. For QCD - there must be less than 16 quarks for it to confine. We know of 6, so it does look like QCD is a confining theory. The weak nuclear force on the other hand does NOT confine because there are too many fermions. Again, all of these arguments are perturbative, so they're not enough for Clay.

There have been studies done on the lattice, where you turn your diffEQ's into finite difference equations and study them on a computer. All the evidence suggests that there is a mass gap (confinement; hadrons), but again, numerical studies are not enough for Clay.

People have come up with "models" that try to work out the structure of hadrons. But the model-dependence is no good - Clay wants something that is completely self-contained. Using only the action of the pure YM, derive the (NON-perturbative) result. Again, no one can do this.

And I love your comment about EFT: many physicists don't take this problem too seriously. But remember that Clay is a mathematics institute - they don't see the problem quite the same way an effective field theorist would. It's a different approach, neither better than the other, just asking different questions.

 

[blechman]

I noticed, in the documentary "the elegant universe" that there were 2 mathematical problems, one was something, and the other was the yang-mills thing. How does this tie in with string theory, does this give the proof.

The two "physics problems" of the Clay institute are the YM mass gap and the navier-stokes equation. Both sound so incredibly straightforward, and yet both are so very hard! I talked already about the YM stuff; the NS problem is to prove that the Navier-Stokes equation, which is meant to describe fluid flow, admits solutions with turbulence. We know experimentally that such solutions must exist (we see turbulence in the atmosphere, oceans, even in our bathtubs!). And yet, no one has been able to construct a complete solution of the equations of fluid dynamics that includes this effect (again, numerical methods and other (physically-reasonable) approximations aside). In fact, it's even worse than that: to claim your money, all you need to do is show that such a solution EXISTS - you don't even need to find it!

Neither of these problems are solved by string theory, although it had its origins in trying to explain confinement, so maybe it will help us with the mass gap. There's been some work in the last decade or so with the "Maldecena Conjecture" which tries to relate strongly coupled YM theories to weakly coupled gravity theories, but it's still a long way away from earning Clay's money!

 

[BenTheMan]

Thus your "proof" of the mass of the electron is somewhat flawed - you've only defined it in a perturbative sense - and the Clay people want a full, nonperturbative solution.

To be fair, I didn't intend it as a proof :) But thank you for explaining these things to me.

But remember that Clay is a mathematics institute - they don't see the problem quite the same way an effective field theorist would. It's a different approach, neither better than the other, just asking different questions.

Yeah---I always took a live and let live approach about these things, but it is funny (to me) to see how vehemently some people adhere to rigor. For example, one of the post docs that I have worked with and I had a very, um, lively (read:he was pissed that I didn't see his point) discussion about the difference between boundary conditions and initial conditions. Imagine if Feynman would have been that way---we never would have had path integrals, which are the only way to do calculations.

 

[blechman]

To be fair, I didn't intend it as a proof :) But thank you for explaining these things to me.

of course. I didn't mean to imply you're sloppy. ;-)

Yeah---I always took a live and let live approach about these things, but it is funny (to me) to see how vehemently some people adhere to rigor. For example, one of the post docs that I have worked with and I had a very, um, lively (read:he was pissed that I didn't see his point) discussion about the difference between boundary conditions and initial conditions. Imagine if Feynman would have been that way---we never would have had path integrals, which are the only way to do calculations.

Rigor in its place. Feynman might have been cavalier about coming up with path integrals (and renormalization, and so many other things), but over time, we have managed to define path integrals (and renormalization, etc) quite rigorously. Sloppiness is tolerable in the beginning, but as you develop more of an intuition for what you're doing, you have to be more careful. Physics is fraught with crazy stories of how being cavalier with your assumptions has led to outright failures in understanding: subtle issues that seem "obvious" at first, and turn out to be completely wrong! Some of these issues are still alive in today's papers.

I don't know what you and your postdoc colleague were arguing about (if you would like to go into it, may i suggest starting another thread - this one seems used up!), but there is something to be said for solidifying one's "fundamental understanding" of the details. BC and IC are not the same things. In the context of GR, for example, such subtle differences could become very important - not only are they different physically, but they're implemented differently in the math! Again, I couldn't tell you more without details, but let's save that for another day.

That aside, let me wrap this up by just pointing out once again that Clay, as a math institute, is interested in math, not physics. These two problems are math problems with their origins in physics. Call it "mathematical physics" if you will; or maybe "physical mathematics" would be even better!