There’s a potentially important new paper on the arXiv from Terry Tomboulis, entitled Confinement for all values of the coupling in four-dimensional SU(2) gauge theory. Tomboulis claims to prove that SU(2) lattice gauge theory has confining behavior (area law fall off of Wilson loops at large distances) for all values of the coupling at the scale of the cutoff, no matter how small. This conjectured behavior is something that quite a few people tried to prove during the late seventies and eighties, without success. Tomboulis is one of the few people who has kept seriously working on the problem, and it looks like he may have finally gotten there. The method he is using goes back to work of ‘t Hooft in the late 1970s, and involves considering the ratio of the partition function with an external flux in the center of SU(2) and the partition function with no such flux. For a recent review article about this whole line of thinking by Jeff Greensite, see here. For shorter, less technical articles by Tomboulis about earlier results in the program he has been pursuing, see here, here, and here.

As far as I can tell, even if this holds up, it won’t get Tomboulis the million-dollar Clay prize being offered for the solution of the Yang-Mills Millenium Prize Problem. There the problem is stated as rigorously constructing Yang-Mills for any gauge group, not just SU(2), showing that it has the expected properties, and a mass gap. I don’t know whether Tomboulis’s methods can prove that there is a mass gap. In any case, I also don’t know why the Clay prize asks for a mass gap rather than confinement, which seems more physically relevant.

The latest HEPAP meeting was this past weekend in Washington and presentations are available here. The news from the DOE is that it looks like Congress will approve a FY 2008 HEP budget at or above the White House request of $782.2 million, an increase of at least 4%. This will include $60 million for ILC R and D (up from $42 million in FY2007). This will include funds for site surveys in the US, which in principle will include sites other than Fermilab, although I find it hard to believe a US site other than Fermilab would end up being chosen.

There’s a report from the University Research subpanel that puts near the top of its recommendations “A higher priority in the overall HEP program should be given to funding directed at university-based theoretical particle physics for the purpose of increasing the number of HEP-grant supported graduate students.” Given the continuing high ratio between students getting Ph.Ds in particle theory and jobs for them when they get out, I would have thought the priority would go to finding ways to fund jobs for students once they get out, rather than increasing the number of Ph.Ds.

There’s a recommendation from the P5 panel that the Tevatron should definitely run through FY2009, and in September they’ll start looking at the case for running even longer than that. There’s also a discussion of “European reaction” to comments from the DOE’s Ray Ohrbach that the US needs a plan in case the ILC doesn’t start happening soon. Typical reaction from the Europeans was said to be that the US is not a reliable partner, and makes unilateral decisions without consultation. CERN has recently been promised a budget supplement over the next few years that could pay for an LHC upgrade (they were in hock over the LHC), and by 2016 that would be done and paid for. At that time CERN will have money to spend on a new big project and a higher energy linear collider using CLIC technology would be a possibility (the semi-joke made was that the work advancing this possibility is now going on mainly in the US, at SLAC).

A progress report from the Fermilab Steering Group, which is supposed to report to the Director on Aug. 1, included extensive discussion of “Project X”, a proposal to dramatically increase the power of proton beams by building an 8 GeV proton linac. Part of the idea is that building this smaller linac would help get experience needed for doing the ILC.

Also at Fermilab, I noticed that they have a web-site devoted to the history of the place, and here is a recent talk by Adrienne Kolb on the subject.

According to CNN, one of the “Geniuses who will change your life” is Harvard’s (soon to be Princeton’s) Nima Arkani-Hamed, who is described as follows:

Nima Arkani-Hamed thinks big. He has a theory that our universe is one of an infinite number of universes — meaning the largest thing we can wrap our minds around is actually pretty tiny

He didn’t pull the “multiverse” out of thin air, though. After becoming a Harvard professor at age 30, Arkani-Hamed first made a name for himself by suggesting that our universe is five-dimensional. Then he moved on to the multiverse, theorizing that our own universe has a hidden feature called “split supersymmetry,” which means that half of all particles have partner particles.

The theory will be tested soon in Switzerland’s brand-new Large Hadron Collider (LHC), and if the LHC finds Arkani-Hamed’s partner particles, it could prove that the multiverse is real — and that our place in it is that much smaller.

The claim that if you see split supersymmetry, this proves that the multiverse is real seems pretty much laughable to me.

Clifford Johnson is in Aspen, at what its director describes as a “summer camp for physicists”. He has a posting about his experiences there, describing how the big topic of conversation is the recent Strings 2007 conference. He asks for people to write in telling about which of the Strings 2007 talks they found most interesting (although so far, a half day later, no one seems to have taken him up on this). Like Jacques Distler, after Witten’s talk, Clifford seems to be most impressed by Seiberg’s talk on BMSSM (Beyond the Minimal Supersymmetric Standard Model) physics, based on his recent paper with Dine and Thomas. The idea is that, even with its 105 extra parameters, the MSSM still requires a lot of fine-tuning to get around the LEP bound on the Higgs mass, so you should analyze adding even more terms to the Standard Model besides the minimal supersymmetric ones. Distler was very excited by the Seiberg talk, arguing that these results should have been found years ago but weren’t. Various people wrote into his blog to point out to him that this wasn’t right, that mostly these things had been done a while ago, including by several authors back in 2003 (see hep-ph/0301121 and hep-ph/0310137). Seiberg et. al. issued a revised version of their paper where they added quite a few references, including one to work back in 1999 by Alessandro Strumia (hep-ph/9906266) where he already discussed the impact of the two operators identified by Dine et. al on the Higgs mass.

**Update**: The debate over string theory really has reached the general public, at least in New York: a few months ago there was a cartoon in the New Yorker, this week, it has made the New York Magazine web-site celebrity coverage.

Peter,

As you say, the string tension (which is experimentally known through

Regge trajectories) is more significant than the mass gap, or

lightest-glueball mass (light glueballs have large widths, can mix with

multi-quark states and so the experimental situation is not as clear).

Maybe the reason the Clay foundation set things up the way they did, is that most people suspected the gap would be easier to prove than confinement. If Terry Tomboulis is correct, however, extending the method to proving a gap is probably not out of the question. He uses a real-space renormalization group argument to compare quantities at strong bare coupling to those at weak bare coupling. The quantity he studies isn’t the string tension, but another quantity, essentially the vortex condensate, which he uses to bound the string tension.

I am spending time reading the paper carefully…

I guess the reason why more hep phd fellowships are necessary

is because we may need, in the near future, even more of these data analysis slaves working for ongoing/upcoming experiments.

Or maybe we are just short of new permanent positions and, since it’s so difficult to defend our area’s research merit these days, it’s much easier to appeal to ‘education purposes’ (what a dragon hunter does? on the very least, he teaches others to hunt dragons!)

Whatever it happens to most of these young physicists after phd is not of much interest; they may spend part of their most creative years with an illusion of a life in physics, but the fact is that, from the very beginning, there would never be enough places for all, just a bright tiny minority. It’s a sad truth in these days of high specialization and changing technology, but these physicists are disposable. They won’t be hired, they will be replaced by new, cheaper, phd students.

“Nima Arkani-Hamed thinks big. He has a theory that our universe is one of an infinite number of universes”

In which part of the landscale did that happen?

“Arkani-Hamed first made a name for himself by suggesting that our universe is five-dimensional.”

They forgot to add he also discovered 3+1 and time travel.

“Then he moved on to the multiverse, theorizing that our own universe has a hidden feature called “split supersymmetry,” which means that half of all particles have partner particles.”

He’s been busy, he also came out with SUSY, SUGRA, the Heterotic string and last episode of ‘numbers’.

“The theory will be tested soon in Switzerland’s brand-new Large Hadron Collider (LHC), and if the LHC finds Arkani-Hamed’s partner particles”,

Are we looking for the Arkino too?

“it could prove that the multiverse is real — and that our place in it is that much smaller.”

This is too complicated to even attempt analysis…

Clearly the CNN people is an example that the reporter’s brain is ‘much smaller’ than we would like it to be…fortunately there is a universe out there where Physics is financed by Ted Turner… 😉

BRITNEY: Forget Peter Woit. Forget Lee Smolin. They don’t know sh*t. I’ve done the calculations. I have a unique compactification that gets all the correct gauge groups and fermion masses. Yeah, yeah … I know. Don’t worry – I’ll post my results on Arxiv later this month!

“In any case, I also don’t know why the Clay prize asks for a mass gap rather than confinement, which seems more physically relevant.”

If I remember rightly, existence of mass gap implies confinement but the converse is not true. So mass gap existence is the stronger statement. It also seems like a perfectly physical statement – it means there are no massless physical states in QCD (in contrast to QED where there are massless photons). A simple intuitive way to see why mass gap implies confinement is this: If gluons were not confined then we would have free massless gluons in the physical QCD spectrum, in contradicton with the existence of a mass gap (i.e. nonzero lower bound on the physical energy spectrum). Same thing for massless quarks.

(Well that’s how I understand it at any rate; if I’ve got something wrong then hopefully someone will correct me.)

It seems Europe was also reacting to Orbach’s statement: (quoted in Wormser’s report)

hello,

thanks for signaling my paper hep-ph/9906266. It has no collaborators because nobody was interested in its main topic: testing a model of extra dimensions at the weak scale with precision data. While completing it, I started to believe that experimental bounds make extra dimensions an unplausible solution to the hierarchy problem, and furthermore another paper appeared with a similar analysis.

At this point, instead of being wise and trashing my little paper, I spent a week-end trying to improve it: adding the aside remark about how non-renormalizable operators raise the MSSM Higgs mass, and writing the introduction in a ironic mood, e.g. “string theory is currently an example of a theory with no parameters that makes no calculable predictions”.

I hope that having anticipated something of Strings 2007 and something of anti-Strings 2007 can be considered as good irony.

If the gauge field is massless, one can calculate and see that the wilson loop satify the perimeter law. So masslessness implies lack of confinement. The converse then must be true. I agree that this is not the same thing as actually calculating the gap.

Ah, the CNN is speaking.

David and Amused,

It isn’t quite that simple.

Gauge fields which are coupled to adjoint Higgs fields have no massless excitations, but do not confine.

Confinement implies a mass scale, but that isn’t quite the same

as a (mass) gap between the ground state and first excited state.

There is an example of confinement with massless particles. This is

the case for a version of compact QED in three dimensions (the reason is that this theory has two mass scales, which should not be the case for QCD in four dimensions. One can tune the gap to be zero, for

fixed string tension, by choosing one of these scales).

On the other hand, any method yielding confinement ought to

powerful enough to settle the issue of the mass gap.

I agree it’s not so simple as that, Peter O.

But the counter example you give is that mass gap does not imply confinement. My point is that since absence of mass implies absence of confinement, confinement should imply mass gap.

This of course does not mean that mass gap implies confinement. So I do not think the Higgs case is a counter example.

To summarize, since masslessness implies the perimeter law, presence of a mass gap is a necessary but not sufficient condition for confinement.

The case of 3 D or 2 + 1 D compact QED is interesting. What are the two mass scales? The charge has a mass scale and there is also the cutoff. If we work on a lattice the cutoff is the lattice spacing. If we work on the continuum and obtain compact QED by spontaneously breaking and SU(2)

gauge theory with the helping hand from a Higgs field, the cutoff is the symmetry breaking scale. So I agree that 2 + 1 D has two mass scales.

What I do not agree with is that 2 + 1 D compact QED exhibits confinement with massless photons. First go to Polyakov’s 1977 Nuclear Physics B paper or his book “Gauge Fields and Strings” and you see that confinement is accompanied by mass generation for the photon.

That the photon cannot be massless if there is confinement can also be understood in simple terms from Wilson loop calculations. I will use latex notation and hope is not too confusing.

The Wilson loop is

\langle \exp{\oint_C A_\mu (x) ds^\mu} \rangle =

\exp{ \oint_C \oint_C D_{\mu \nu} (x – x’) ds^\mu ds’^\nu }

where D_{\mu \nu} (x – x’) is the gauge field propagator which is gauge variant of course. The Wilson loop is gauge invariant because the integrations over closed paths lead to zero for the gauge variant parts of the propagator.

Now to the point, if we use the massless propagator we do not get an area law for the Wilson loop in 2 + 1 D (the same thing happens in 3 + 1 D ). Hence if the photon is massless there is no confinement.

So even though 2 + 1 D has two mass scales it is not true that one of them can be tuned to have confinement and a massless photon. It is possible that one of the mass scales can be tuned so that the photon becomes massless, but at such a point the theory does not confine.

David, I don’t understand your reasoning at all. The gauge fields in QCD are necessarily massless – a mass term would break gauge invariance. However, the massless quanta of the gauge fields – the gluons – are not physical states of the QCD Hamiltonian (in contrast to photons in QED) since they don’t satisfy the nonabelian Gauss Law constraints (they aren’t color singlets). The question that then arises is: can one have color singlet states consisting of a number of essentially non-interacting free, deconfined gluons? Such a state would be massless (since the individual gluons are massless), implying that QCD does not have a mass gap. But there is no sign of such states in experimental low energy hadronic physics (which is where we would see them if they existed). Lattice studies indicate that color singlet purely gluonic states do exist in QCD – the so-called glueballs – but they are massive (quite heavy in fact) and have yet to be seen in experiments afaik.

The lightest state in hadronic physics is the pion, which definitely has nonzero mass, so if QCD is to describe hadronic physics like it is supposed to then mass gap is implied by experiment. This makes it very natural that proof of mass gap should be a criterion for the Clay prize.

Peter O., my post crossed with yours, I’ll go back and read what you wrote now.

To summarize if A implies B, then it is true that not B implies not A.

Masslessness implies no confinement, thus confinement implies mass gap.

I understand there are nontrivial issues with extracting certain information from a non-Abelian gauge theory and this is an understatement. And calculating the gap might be a herculean task. But as long as the lack of mass means no confinement, it seems that confinement implies a mass gap.

On the other hand like I said and Peter O.’s Higgs phase example shows perfectly, the presence of a mass gap certainly does not imply confinement!

Peter O.

I would love to hear your take on the paper. It’s strengths and its weaknesses or open questions.

If I understand correctly, Peter O.’ counter-example to “mass gap implies confinement” involves spontaneous breaking of gauge symmetry, resulting in massive gauge field. In that case the intuitive argument I gave in my first post breaks down. Is there a counter-example where gauge symmetry is not broken (as in QCD)?

Btw, the “if I remember rightly, mass gap implies confinement” in my earlier comment was referring something I once heard Pierre van Baal say (in reference to QCD). It is possible I might be remembering wrongly though…

David, I will need to go and read up on these things again to check your claim that massless gauge field implies perimeter law (it’s been a while..). Did Wilson really require massive gauge field when he derived the area law in strong coupling lattice QCD? I don’t remember that!

amused,

As I said I have never stated that mass gap implies confinement as that is explicitly false.

What is not false is that confinement implies mass gap because masslessness

implies lack of confinement.

Once again A implies B is equivalent to not B implies not A, but does not say anything about not A implying not B, and in fact this last statement could very well be false.

No Wilson did not require a massive gauge field directly. But in his paper he did calculate that a massless gauge field implies no confinement. Therefore on can conclude that confinement implies mass gap.

Mass gap certainly does not imply confinement and a very good counter example is the “spontaneous breaking” of gauge symmetry mentioned by

Peter O. where there is a mass gap but no confinement.

David,

I wrote a paper when a dumb grad student, publ. in

Nucl. Phys. 1982, where it is shown that in the

STANDARD continuum limit of 3D CQED, the mass

gap vanishes, with fixed string tension. This is

because there are two scales, namely the coupling constant

and the monopole fugacity. If one changes the monopole

fugacity, it is possible to keep the gap in the continuum

limit (though I did not mention this in the paper). Appropos of

nothing, the paper is mainly about generalized gauge theories of

p-form fields, their solitonic p-branes, duality and confinement

phases. Anyway, the masslessness of the (confining) continuum limit

was shortly therafter proved rigorously by Gopfert and Mack.

Now presumably there is only one scale in QCD, so this should not

happen. The question is what standard of rigor you demand. It is

not utterly inconceivable though that the gap could vanish

(with all other excitations massive), though I would be greatly

surprised if this happens.

Now essentially what Tomboulis does is show that one can

run the bare coupling all the way to the strong-coupling

region. If his paper is right, there is “no doubt” there is

a gap – I mean that I would be convinced, but that’s

not yet a proof. What would be interesting is an estimate

of the gap (he does have an estimate of the string tension).

As far as the question of the paper being right is concerned, I don’t have an opinion yet. I am spending all my spare time trying to figure it out. Right now I am a little bogged down in how Tomboulis

understands and uses reflection positivity.

Peter O.,

I could buy what you say and I’ll definitely check out your paper.

But I would like to know how do you scape the Wilson loop argument. This confuses me. If you calculate the Wilson loop with a massless propagator in 2 + 1 D and 3 + 1 D, one does not obtain an area law. I would take this to mean no confinement. So what gives?

Is it that we have confinement without an Area law? How does this come about?

Is it an order of limits issue because the point at which the gauge fields just goes massless is critical point and one needs to be very careful?

David,

“As I said I have never stated that mass gap implies confinement..”

No, it was me who stated that.

“Once again A implies B is equivalent to not B implies not A, but does not say anything about not A implying not B, and in fact this last statement could very well be false.”

I think we can agree on that.

“No Wilson did not require a massive gauge field directly. But in his paper he did calculate that a massless gauge field implies no confinement.”

Now I’m really confused. I was under the impression that Wilson derived the area law using the “Wilson action” describing massless gauge fields, and without sneaking in a mass for the gauge field anywhere along the way. Is this correct or is it not correct (yes or no please).

“Mass gap certainly does not imply confinement and a very good counter example is the “spontaneous breaking” of gauge symmetry mentioned by Peter O. …”

I’m sure it is an excellent example, but unfortunately not so relevant for the situation I happen to be interested in which is where gauge symmetry is not broken (as in QCD).

“But I would like to know how do you scape the Wilson loop argument. This confuses me. If you calculate the Wilson loop with a massless propagator in 2 + 1 D and 3 + 1 D, one does not obtain an area law. I would take this to mean no confinement. So what gives?”

OK, David, I think you are assuming that the Wilson-loop can be

reliably computed by gluon emission and absorbtion by a source. This is what perturbation theory tells us (there are of course corrections of more exchanges and higher loops), but it is very misleading for QCD, where gluon exchange is only meaningful at short distances. The contribution for large separations on the loop is not given by a finite number of gluon exchanges (in fact, most of the serious people in this field think of gluon exchange as a red herring).

In contrast, strong-coupling expansions don’t give any gluons, just

glueballs. The problem with these expansions is that there is no

reason to assume they don’t have a finite radius of convergence. They

are at best a caricature of the continuum limit, and they tell us little

about high-momentum physics.

amused, if you go to Wilson’s paper he calculates his famous loop with massless gauge fields and shows that it does not satisfy the area law.

The bare fields are massless; you are correct in this. The question is whether the long distance renormalized gauge fields are massless. For the this last case there certainly is no confinement.

He did this to show that QED for weak coupling which has massless gauge fields does not confine.

He then goes on to argue strongly, not show, but good enough to convince me. that for strong coupling the Wilson loop satisfies the area law and does there is confinement. He did not calculate the mass gap nor explicitly state

that it ha gap, but given that masslessness implies no confinement it is certainly implicit.

Now, Peter O. tells me that one can show that 2 + 1 D CQED confines without a mass. Like I told him I could take his word on this, but it baffles me since one can calculate the Wilson loop for a massless gauge field

in 3 + 1 D and 2 + 1 D and it does not satisfy the area law. So it looks that the has a case where there is confinement without and area law.

How can this happen, I do not understand. I would love for Peter O. to clarify as I would learn something new. I can buy it but I remain sceptical until I understand it better.

Does this help amused?

Peter O., Thanks

Yes I agree that going to the strong coupling expansion is non trivial and in a a sense I was using perturbative thinking

Amused,

Wilson’s original paper did strong-coupling expansions for

pure gauge theories. In these expansions there is a gap

and confinement. It isn’t that one implies the other, but

that both are consequences of the the theory being

weakly-correlated.

One question Peter O.

In your work on confining massless 2 + 1 CQED. Are the massless excitations gauge fields? I would think not, but I am not sure.

David,

That’s a good question. From what I remember, they are just

photons – the correlation function of the field strength

looks just like ordinary free QED.

David, thanks for your efforts to explain. I will need to go take a look at Wilson’s paper again to remember how this all hangs together. One immediate objection I have to your argument is that the renormalized gauge fields could acquire mass. Surely gauge invariance (Ward Identities) prevents this?

Anyway, it is well past my bedtime here on the other side of the world, so I’ll have to take a break from the discussion for the time being.

Yes amused, Ward identities prevent this. I misspoke by saying that the aguge fields have mass. It is thought that the spectrum is fully gapped. So what is believed is that the gauge fields disappear from the spectrum, but you have things like Gluballs that have mass so the spectrum is gapped.

This is believed for many reasonable arguments and I buy them. But it has never been proven even semi rigorously I believe.

In Wilson paper all of this does not hang fully together. This beliefs and the reasonable arguments that point to them was built slowly through the mid 70’s to the early 80’s both from lattice gauge theory and from continium QFT.

It’s pretty amusing how people like Distler get all excited over something when a celebrity like Seiberg says it, but not so much when a nobody said it years before. Same thing happened with the anthropic landscape. Modern physics is just a fashioin show.

David,

I think if you read Wilson’s paper carefully, it is quite sensible. He

was very careful.

Peter O. and anyone else,

I’m pretty sure I remember hearing from a very authoritative source (P. van B.) that mass gap implies confinement in QCD. If this is true it would explain why the Clay prize asks for proof of mass gap but doesn’t mention confinement. Can anyone out there confirm this?

David,

what you wrote in your last comment seems to be converging to what I had written earlier. Is there anything we still disagree on?

Has anybody here actually read the Witten-Jaffe text?

I just took a look at it now (the article is

here). No mention there about mass gap implying confinement (or the converse)… A fundamental physical aspect of the mass gap that they emphasize is that it implies YM interactions are short-range (which they must be if QCD is to describe the strong nuclear force).

amused,

I don’t know what Pierre’s argument was. There is no rigorous

proof or estimate of the gap from the assumption of confinement

(that I am aware of, and I have worked on this, on and off for a long

time). There are lots of physical argument; I gave one above,

assuming there is one scale. One expects that $K=M^{2}/\sigma$ is

universal for this reason, where $\sigma$ is the string tension and

$M$ is the gap. So can $K$ be zero? I say certainly not, but my statement is not a calculation or an estimate of $K$.

By the way, Terry’s argument, if it is right, does imply a mass

gap for another field theory, namely the principal chiral SU(2)

1+1-dimensional nonlinear sigma model.

In Wilson’s original paper if you go to the last paragraph of the second column of page 2447 and about two thirds of the first column of of page 2448. Wilson calculates his loop for a massless gauge field propagator in 4D (or 3+1 D) and shows that it satisfies the perimeter law.

Therefore massless gauge field leads to perimeter law, that is no confinement. At least in 4D masslessness implies no confinement and thus confinement implies mass gap. Now as amused said if is the gauge field

that acquires the mass there seems to be conflict with Ward identities. A possible resolution is that the gauge fields disappear from the spectrum and what is gapped are objects like glueballs, but no ones know for sure

Now mass gap in gauge theory does not imply confinment. The Higgs phase is a counter example. Now it is believed that if there is no symmetry breaking and there is mass gap in a gauge theory it has to confine. There are reasonable arguments for this, but no one knows for sure.

So amused, I think I agree mostly with you except on the statement that a mass gap necessarily implies confinement. Although I confinement probably implies a mass gap. Peter O. said that he has a counter example in his early paper. I have not gotten around to reading and studying the paper.

“In Wilson’s original paper if you go to the last paragraph of the second column of page 2447 and about two thirds of the first column of of page 2448. Wilson calculates his loop for a massless gauge field propagator in 4D (or 3+1 D) and shows that it satisfies the perimeter law.”

David,

This is assuming one can calculate the loop by gluon exchange. Wilson is just pointing out that perturbation theory won’t work,

not that a gap is necessary for confinement.

If you use a massive propagator you will also find a perimenter law.

Peter O.

Wilson was pointing out that a massless propagator leads to a perimeter law.

I agree that if the theory is strongly coupled there are important corrections that might invalidate this thinking. But I would say that these corrections

would have to violently change the nature of the gauge field propagator

so that it does not correspond to propagation of a massless excitation.

The interpretation of the gap is my reading, but as I just said I agree that nonperturbative effects could invalidate this thinking. So I am not fully dogmatic and this reading could be wrong. I’ll check out your paper too. I have just been caught up with other things.

Peter O. I agree with your comment of the massive propagator.

OK David, I’d be interested in any other comments you have.

Thanks for the replies Peter O. and David. I guess Pierre must have had some physical/heuristic argument in mind like you said. (Or it could be that I’m just misremembering what he said, in which case I’ll be glad that I’m anonymous if he ever sees this!)

Peter Orland said:

That’s the way I’ve got it figured. Connes, Jaffe, Wiles and Witten

were looking for problems with high probablity of solution. Just look at their their past behavior! Wiles picked Fermat’s Last

Theorem for its tractability, while Connes and Witten pursued the low risk roads of Noncommutative Geometry and String Theory.

The Poincare Conjecture is another good example. Heck, the writing was on the wall before that Perelman character showed up.

As for the Riemann Hypothesis, a string theorist said it best to Michael Berry:

Special note to Asperger sufferers: Try not to take this message too seriously.

Neville,

It’s not that I don’t appreciate your light-hearted remarks, but both

problems – confinement and the gap – were always regarded as

super-tough. I can’t read anyone’s mind, but I was just trying to

guess the Clay F.’s motivation.

Somehow out of topic, and maybe not on the various news theme:

I found there is a loops 07′ conference also in Czech republic, Lubos’ land, in Charles U. indeed…

http://www.karlin.mff.cuni.cz/~loops07/

I am sure he will find this rather amusing too 😉

No idea how loops are related to Agriculture though… 😀

Just a few final thoughts on mass gap vs. confinement in QCD (sorry I can’t resist). A mass gap certainly implies that any particles corresponding to massless excitations of the fields must be confined in some way, since otherwise there would be states describing free massless particles in the physical spectrum. In particular, gluons must be confined; this ties in with the point emphasized by Jaffe & Witten that mass gap implies that the YM interactions are short-range. Similarly, massless quarks must also be confined. On the other hand, this reasoning does not imply confinement of massive quarks. I guess one can invoke a scale argument in this case, as Peter O. did, to see why confinement is expected, but it is of course not a proof.

Actually, for QCD to have a mass gap the quarks must be massive, since otherwise there would be massless Goldstone bosons associated with the spontaneous chiral symm. breaking (i.e. the mesons would be massless).

From looking at the Jaffe-Witten article it seems there are 3 reasons why mass gap rather than confinement was chosen for the Clay prize problem:

(1) Its physical significance: implies that YM interactions are short-range.

(2) Simpler to formulate than confinement: it just means the physical energy spectrum has a strictly positive lower bound.

(3) Potential mathematical significance: since mass gap implies that the interactions are short-range, it makes it potentially easier to extend results from YM on R^4 to YM on general 4-dimensional spacetime manifolds.

Amused,

Confinement doesn’t just mean that particles disappear from the spectrum. It is COLOR confinement. The excitations which do appear

are color singlets. There are no asymptotic particle states with color.

Peter Orland says:

I can’t read anyone’s mind, but I was just trying to

guess the Clay F.’s motivation

I suspect this may be right. In a conversation I’ve had with one of the problem’s formulators (although we weren’t specifically discussing the mass gap vs. confinement issue), he said that they tried to give the simplest formulation which would prove the type of result he wanted.

Peter O,

I know that. But without knowing about QCD dynamics one could envisage color-singlet states describing a collection of deconfined gluons (as I mentioned in an earler comment). Something like a very dilute, spread out glueball. Such a state would have effectively zero mass, and is therefore excluded by mass gap.

Anonymous,

“Simplest formulation” isn’t the same as “easier to prove”, which was what Peter O suggested. But yes, as mentioned above, I also got the impression from the Jaffe-Witten article that a reason they chose mass gap rather than confinement was because the problem was simpler to formulate.

Whoops, just realized that it isn’t possible to have states which are both color-singlet and describe a collection of free gluons or quarks at the same time. (There would have to be gluon strings between the quarks, or closed gluon loops, etc…). So I take back what I wrote earlier, sorry ’bout that.