I have added pointer to:

- Michael Atiyah,
*The millennium problems II*, talk at CMI Millennium Event at Collège de France (24-25 May 2000) [YT, discussion of Yang-Mills&MassGap starting at: 29:40]

together with the following transcript of the last couple of minutes of the part on the mass gap problem (maybe this should eventually go to a separate entry, for the moment I have dropped it here):

]]>40:20 What is the future of this problem [Yang-Mills & mass-gap] in physics, what could happen?

40:24 This is now my personal guess or, if you like, hints [or] suggestions.

40:30 You can try a direct analytical assault. If you are a young man and brave and you have a long way into thre future, you can just get out your hammer and chisel and crack away.

40:47 Or you might think that you want something different.

40:50 You might find to get a better understanding of the formal structure of quantum field theory of this Yang-Mills type.

40:58 These Yang-Mills theories have very elaborate mathematical formal structure, which incorporates very mysterious dualities of various kinds and also a thing called

supersymmetry, and many of the applications in mathematics incorporate all these.41:51 So here there is a very elaborate structure, and it is just possible that if we get a better understanding of that structure then that might have given us a better way of trying to lay the foundations.

41:25 For example, a problem that can be described in two different ways, by duality, two dual pictures might look quite different, one of them might be practical and the other one might be intractable.

41:35 They are certainly not at all unreasonable to think this might be a way starting to understand better these formal operators[?], the formal operators here is very sophisticated.

41:47 You could of course sit back and wait, await development in string theory.

41:54 String theory and the latest things that follow string theory, called M-theory and every other new theory, develop [at] an enormous rate.

42:00 The theoretical physics community here is tremendously active, there are very beautiful things happening every week, new results come out, many of them will have mathematical overtones.

42:10 And it could be that if we wait a few years there’ll be such a totally new picture emerging from string theory that we’ll get a better idea how to go about attacking the problem, so that’s not an unreasonable expectation.

42:21 You see, trying to lay the foundations for physics is like an architect trying to lay the foundations for a building that’s rapidly going up.

42:30 I mean, which do you do first, it’s not quite clear: You want to see the design or you want to lay the foundations. Maybe you want the design first.

42:39 Then, of course, you might also sit back and say: Well, quantum electrodynamics was very hard, because we didn’t incorporate extra things like proton and neutrons, we only worried about electrons. Maybe even that isn’t enough perhaps we should have incorporated gravity, then we’ll have all the forces under control and perhaps at that stage the fundamental theory will be even easier.

43:00 People in string theory aim here and this is the ultimate theory, and perhaps the ultimate theory will be easier than the transitional theories. But who knows.

43:09 The big challenge [for] mathematics and the physicists of the 21st century is to really make progress along this program by whatever method they can. Good luck to you.

also saved the pdf-s of these two items, whose links had rotted, too, by recovering them via the WaybackMachine and then uploading them to the nLab-server:

Arthur Jaffe, Edward Witten,

*Quantum Yang-Mills theory*(2000) [pdf]Michael Douglas,

*Report on the Status of the Yang-Mills Millennium Prize Problem*(2004) [pdf]

Noticed that the Clay Institute link for the mass gap problem had rotted. I have updated it (www.claymath.org/millennium/yang-mills-the-maths-gap) here and will try to update it elsewhere.

]]>added pointer to today’s

- Alessandra N. Braga, Wagner P. Pires, Jeferson Danilo L. Silva, Danilo T. Alves, Van Sérgio Alves,
*Renormalization of the band gap in 2D materials near an interface between two dielectrics*[arXiv:2306.04759]

added pointer (here) to

- Yakov Shnir, Section 9.1. of:
*Magnetic Monopoles*, Springer 2005 (ISBN:978-3-540-29082-7)

as another (of rather few) examples of nuclear-physics texts that connect the physicist’s confinement problem to the mathematician’s mass gap problem

]]>added pointer to today’s:

- Gergely Gábor Barnaföldi, Vakhtang Gogokhia,
*The Mass Gap Approach to QCD*(arXiv:1904.07748)

I have sub-divided the section “References – Approaches” into “References – Approaches – Computer lattice QFT” (which is what used to be there) and “References – Approaches – Rigorous lattice QFT” (which is new).

The reference given there has more pointers to this and other approached. Eventually these further citations ought to be included here, but I leave it at that for the moment

]]>added this problem description in terms of rigorous lattice QFT:

- Sourav Chatterjee,
*Yang-Mills for probabilists*, in:*Probability and Analysis in Interacting Physical Systems*, PROMS**283**(2019) Springer (arXiv:1803.01950, doi:10.1007/978-3-030-15338-0)

added publication data to

- Craig Roberts,
*On Mass and Matter*, AAPPS Bulletin volume 31, Article number: 6 (2021) (arXiv:2101.08340, doi:10.1007/s43673-021-00005-4)

as well as pointer to footnote 2 and reference [17] in that text (which explicitly refer to the CMI problem)

]]>added publication data for:

- Ludvig Faddeev,
*Mass in Quantum Yang-Mills Theory*, Bull Braz Math Soc, New Series 33(2), 201-212 (arXiv:0911.1013, pdf)

have added one more quote to the list, from today’s

- V. A. Petrov,
*Asymptotic Regimes of Hadron Scattering in QCD*(arXiv:1901.02628)

This is a commonplace that so far we do not have a full-fledged theory of interaction of hadrons, derived from the first principles of QCD and having a regular way of calculating of hadronic amplitudes, especially at high energies and small momentum transfers. The problem is related to a more general problem that QCD Lagrangian would yield colour confinement with massive colourless physical states (hadrons).

s

]]>added one more quote in the above vein:

- Jeff Greensite,
*An Introduction to the Confinement Problem*, Lecture Notes in Physics, Volume 821, 2011 (doi:10.1007/978-3-642-14382-3)

]]>Because of the great importance of the standard model, and the central role it plays in our understanding of particle physics, it is unfortunate that, in one very important respect, we don’t really understand how it works. The problem lies in the sector dealing with the interactions of quarks and gluons, the sector known as Quantum Chromodynamics or QCD. We simply do not know for sure why quarks and gluons, which are the fundamental fields of the theory, don’t show up in the actual spectrum of the theory, as asymptotic particle states. There is wide agreement about what must be happening in high energy particle collisions: the formation of color electric flux tubes among quarks and antiquarks, and the eventual fragmentation of those flux tubes into mesons and baryons, rather than free quarks and gluons. But there is no general agreement about why this is happening, and that limitation exposes our general ignorance about the workings of non-abelian gauge theories in general, and QCD in particular, at large distance scales.

added another good citation in the above vein:

- Brambilla et al..
*QCD and strongly coupled gauge theories: challenges and perspectives*, Eur Phys J C Part Fields. 2014; 74(10): 2981 (doi:10.1140/epjc/s10052-014-2981-5)

]]>The success of the technique does not remove the challenge of understanding the non-perturbative aspects of the theory. The two aspects are deeply intertwined. The Lagrangian of QCD is written in terms of quark and gluon degrees of freedom which become apparent at large energy but remain hidden inside hadrons in the low-energy regime. This confinement property is related to the increase of $\alpha_s$ at low energy, but it has never been demonstrated analytically.

We have clear indications of the confinement of quarks into hadrons from both experiments and lattice QCD. Computations of the heavy quark–antiquark potential, for example, display a linear behavior in the quark–antiquark distance, which cannot be obtained in pure perturbation theory. Indeed the two main characteristics of QCD: confinement and the appearance of nearly massless pseudoscalar mesons, emergent from the spontaneous breaking of chiral symmetry, are non-perturbative phenomena whose precise understanding continues to be a target of research.

Even in the simpler case of gluodynamics in the absence of quarks, we do not have a precise understanding of how a gap in the spectrum is formed and the glueball spectrum is generated.

after all the recent edits to the References here, I only now noticed that the Idea-section of the entry itself was at best a stub. Now I wrote some minimum text there.

]]>Ah, no. But writing up an article on a non-perturbative physics model and trying to put things in proper perspective in the introduction.

]]>I hear a grant application in preparation :-)

]]>Thanks!! And so it’s relatively recent, too (2015). Excellent.

]]>See here for that second one.

]]>Presumably one can hope to find ways of using computers to help calculate the consequences of any kind of theory.

I thought the comparison to the Riemann hypothesis that I suggested above should be helpful to see this: No amount of checking the Riemann hypothesis case-wise on a computer will count as a proof or as providing theoretical understanding. Same for lattice QCD-tests of confinement/mass gap.

Meanwhile I found two quotes of the explicit kind that I am after. Maybe you could help me with the second, for here I only have a pdf-file, which says it is a chapter 7 of some bigger document, but I can’t find that bigger document, or just its citation data.

Here is the first:

- J J Cobos-Martínez,
*Non-perturbative QCD and hadron physics*2016 J. Phys.: Conf. Ser. 761 012036 (doi:10.1088/1742-6596/761/1/012036)

However, the QCD Lagrangian does not by itself explain the data on strongly interacting matter, and it is not clear how the observed bound states, the hadrons, and their properties arise from QCD. Neither confinement nor dynamical chiral symmetry breaking (DCSB) is apparent in QCD’s lagrangian, yet they play a dominant role in determining the observable characteristics of QCD. The physics of strongly interacting matter is governed by emergent phenomena such as these, which can only be elucidated through the use of non-perturbative methods in QCD [4, 5, 6, 7]

And here is the second, of which I am lacking the citation data:

]]>There is overwhelming theoretical and experimental evidence that QCD is the theory of strong interactions. Yet, QCD is to a large extent unsolved. In particular, standard perturbation theory becomes completely unreliable in the infrared regime, where QCD is strongly coupled. Experimentally, there is a large number of facts that lack a detailed qualitative and quantitative explanation. The most spectacular manifestation of our lack of theoretical understanding of QCD is the failure to observe the elementary degrees of freedom, quarks and gluons, as free asymptotic states (color con- finement) and the occurrence, instead, of families of massive mesons and baryons (hadrons) that form approximately linear Regge trajectories in the mass squared. The internal, partonic structure of hadrons, and nucleons in particular, is still largely mysterious. Since protons and neutrons form almost all the visible matter of the Universe, it is of basic importance to explore their static and dynamical properties in terms of quarks and gluons interacting according to QCD dynamics.

That seems quite complicated how terms – perturbative/non-perturbative, effective, phenomenology, computational model – intertwine here and in general.

actual theory as opposed to computer simulation

Presumably one can hope to find ways of using computers to help calculate the consequences of any kind of theory.

Is the issue here more that the right kind of theory is not yet in place,

]]>the explicit non-perturbative formulation of Yang-Mills theories such as QCD is presently wide open

Have read it now. Besides reporting that neat computation, this article is an exceptionally good account of lattice QCD in general.

(While this is nice, let me recall that the point in #1 is really about actual theory as opposed to computer simulation. Might be compared to the situation with the Riemann hypothesis: computer experiment checks the hypothesis already to fantastic “accuracy”, but none of this has any effect on the hypothesis as a theoretical open problem. Same here with confinement/mass gap and its implications.)

]]>Thanks! That’s good.

]]>Maybe this much longer review article by two of the authors of that Science paper should be added:

- Zoltan Fodor, Christian Hoelbling,
*Light Hadron Masses from Lattice QCD*, Rev. Mod. Phys. 84, 449, (arXiv:1203.4789)

So I’ll add it.

]]>Thanks for the pointer to the arXiv article! I’ll include pointer to that into relevant entries.

I hadn’t seen that blog post you point to, but it’s one among many other informal such discussions that I have seen. Am hoping to collect more citeable references.

]]>