There’s a conference at Harvard this week celebrating (somewhat early) the 60th birthday of the geometer Shing-Tung Yau. Since I was passing through Cambridge on the way back from a short vacation, I managed to catch a few of the talks, including two quite nice ones on mathematical physics from Is Singer and Edward Witten.

Singer’s talk was entitled “The Interface between Geometry and Physics, 1967-2007”, and summarized some of the advances in this area that he has been involved with over the years. 1967 was the year of a Battelle conference in Seattle on the intersection of mathematics and physics, organized by deWitt and Wheeler. Singer displayed a copy of a 1966 letter from Feynman to Wheeler turning down an invitation to attend, with the explanation

I am not interested in what today’s mathematicians find interesting.

At the 1967 conference Robert Geroch talked on the topic of singularities in GR, and Singer recalled inviting Geroch to talk at the 1973 geometry conference at Stanford on this topic and the positive mass conjecture. By 1975 Singer had learned from Jim Simons at Stony Brook that non-abelian gauge fields were exactly the connections on principal G-bundles studied by mathematicians. The news of the BPST instanton solution and its significance for physics caused him to seriously start working in this area, work conducted with Hitchin and Atiyah at Oxford. They made use of the index theorem to both calculate the dimension of the moduli space of instantons, and to show that the Dirac operator had zero-modes in instanton backgrounds. Later, Atiyah and Singer interpreted the local gauge anomaly using the farmilies index theorem.

Just as Atiyah did at his recent talk at the Bott conference, Singer interspersed his historical talk with remarks identifying unsolved problems that he thinks are worth attention. The first of these has to do with K-theory, extended to depend not just on vector bundles, but on vector bundles with connection. Here the open problem has to do with the analog of the families index theorem. There’s a topological index (that takes values in the extended K-theory of the family), and the open problem is to define an appropriate analytical index and show topological=analytical equality. Singer had been hoping to have some new results to report on this, but he says that things turned out to be trickier than he had expected, so maybe he’ll have an answer at Yau’s 65th birthday.

The next topic was that of quantum Yang-Mills theory and the Millennium problem of proving the existence of a mass gap. Singer talked mainly about 2+1 YM, his conversations with Feynman about this, and his hopes that the positivity of the sectional curvature of the natural Riemannian connection on the space of gauge potentials modulo gauge transformations could be exploited to prove that there must be a mass gap. Proving this in 2+1d was his second open problem.

His final topic was mirror symmetry and S-duality. Here he speculated that this (and M-theory in general) might have something to do with a phenomenon from operator algebra theory. Unlike for N by N matrices, where all maximal abelian subalgebras of self-adjoint operators are conjugate, for certain C* algebras (rings of operators of type II), there are inequivalent such maximal abelian subalgebras. I gather that his idea is that these might correspond to the existence of lots of inequivalent limits of M-theory.

The next morning Witten gave a talk on quantum Yang-Mills and the Millennium problem, saying that this was in response to a request from Yau to explain at a basic level to a wide mathematical audience what this is about. He gave an extremely lucid explanation of the mass gap problem, taking the Hamiltonian point of view. This supplements what he and Jaffe did in the official write-up of the problem, which deals more with the Euclidean picture. As motivation, he explained in detail what happens in the Abelian case, where one can compute everything and there is no mass gap. The audience was appreciative and I think got something out of this, unlike quite a few talks of physicists to mathematicians, which tend to start at much too high a level of complexity, ensuring that only experts can follow.

Witten avoided one aspect of this problem, the one that most fascinates me, that of how you handle the gauge symmetry. In the Abelian case there are several equivalent ways of doing this, but in the non-Abelian case, at least in the continuum, one needs to understand BRST symmetry non-perturbatively, and this remains a difficult problem, one with deep connections to open problems in mathematics.

Hi Peter,

Thanks for the post. Probably just a typo but calling ‘Is’ ‘Iz’ drives ‘Is’ crazy because it’s hard to stop once it gets going.

I am particulary glad to see discussion about the period of intense activity between geometry and physics before the anomaly cancelation. I believe that for Is the great epiphany was realizing that the BPST instanton was exactly the Hopf fibration. At that moment, I believe he understood immediately (and likely better than anyone else at the time) that the interaction between the two fields was all but destined to be a full fledged revolution. I’ve heard him tell the story two or three times and it is clear to me that he has no self-interest beyond preserving some understanding of his part in the exact chain of events. Yang and Atiyah have also spoken and/or written elloquently on this subject, but I have always found Singer’s role to be particularly interesting.

It is going to become increasingly hard for modern mathematicians and physicists to understand that the major stumbling block was the lack of a single mature example connecting the two fields. How can it be that ‘differential forms’ were once ‘exotic’? How could the quaternionic Hopf fibration be anything other than an excercise for beginning students?

And yet to some of those who made the initial leap, those first steps were more amazing than the unbelievable story which followed.

Also Singer is personally very kind to Bob Hermann on the issue of the Wu-Yang dictionary. While taking nothing away from Simmons, Wu or Yang, he believes that more credit should have found its way to Bob Hermann and his unusual set of books on geometry and physics. Hermann spoke of his ‘probable mistake’ in leaving tradional forms of peer reviewed research behind. I myself felf after meeting the man after a lecture I gave at MIT that he simply had no choice. He was simply a man ahead of his time and a marvelous case study about where peer review fails abjectly.

I have always thought Singer a mensch for going out of his way to make sure that everyone involved got full credit for his role in this revolution. If only all history was written by such winners we would have a far better road map for the revolutions to come.

History has been very kind to Singer but, at least in my opinion, not yet kind enough.

Best,

Eric

Eric,

Could you please elaborate on the Robert Hermann aspect of the story?

Eric,

Thanks, fixed Singer’s name. Thanks also for mentioning Hermann. His books are remarkably ahead of their time.

Mathphys,

For example, see Hermann’s books “Vector Bundles in Mathematical Physics”, published in 1970. I’ve always assumed that there were other mathematical physicists well-aware of the gauge theory/connections on bundles dictionary before this became well-known after 1975 (if only from reading Hermann’s books…). This only started getting attention though after the Standard Model came together in 1973, and people started thinking about non-perturbative questions in it. The ‘t Hooft-Polyakov monopole in 1974 and the BPST instanton in 1975 were the crucial examples that got a lot of people interested in this.

I’d like to concur on how wonderful Witten’s talk was. As a nonexpert who normally gets left in the dust in less than 5 minutes of any talk on quantum field theory, I was amazed and delighted that I either understood or had the illusion that I understood Witten’s talk from beginning to end. Naturally, that means that Witten successfully hid all of the reasons why the mass gap conjecture is so difficult. But I was able to gain a reasonable appreciation of what the goal is.

The conference itself is quite overwhelming (I left yesterday with 2 and half days to go). At first, I could not understand why anyone would want to schedule so many talks (about 50, I think) in such a short time. But then I finally realized that Yau himself has no trouble processing this amount of information; he is sitting in front and clearly understanding every talk. But many of us mortals are being left in the dust.

And I also agree that it is nice to hear Hermann’s name mentioned after all these years. When I was a graduate student learning differential geometry and interested in physics, Hermann’s books were fascinating, if maddening. They provided a reasonable guide to modern differential geometry, but I could never understand his explanation of the connections to modern physics.

With respect to Robert Hermann as “… a man ahead of his time and a marvelous case study about where peer review fails abjectly …”

consider these quotes from two letters he published in his 1979 book “Cartanian Geometry, Nonlinear Waves, and Control Theory: Part B”:

“… I am not the only one who has been viciously cut down because I tried to break out of the rigid shell and narrow grooves of American mathematics. … My proposal was to continue my … work with … Frank Estabrook and Hugo Wahlquist of the Jet Propulsion Laboratory. … I most deeply resent the arrogance of the referee #3 toward their work … typical … arrogance of Referee #3 is his blather about the “prematureness” of our work … Now, we are working in a field – nonlinear waves – which is moving extremely rapidly and which has the potential for the most important applications, ranging from … Josephson junction to … fusion … and I am supposed to sit back and wait for Professor Whosits to tell me when he thinks problems are “mature”…

I sent the papers he mentions to very few people … I am also interested to note that he did look at them, since there is considerable overlap in methodology with a recent paper by one of his students, with no mention of my papers in his bibliography …

any money spent by NSF on a Mathematics Research Institute would be down the proverbial rat hole – it would only serve to raise Professor Whosits’ salary and make him ever more arrogant. It would do more good to throw the money off the Empire State Building: at least there is a chance it would be picked up and used creatively by a poor, unemployed mathematician …

This issue transends my own personal situation …

Most perversely, the peer review system … works as a sort of Gallup poll to veto efforts by determined individuals … As budgets have tightened, the specialists fight more and more fiercely to keep what little money is available for their own interests. Thus, people with a generalist bent are driven out …”.

My understanding is R Hermann was a professor at Rutgers in the 70’s. Did he quit to devote himself to writing his books?

MathPhys asks “… Did … R Hermann … quit … Rutgers … to devote himself to writing his books? ..”.

Hermann said in letters published in his 1979 book “Cartanian Geometry, Nonlinear Waves, and Control Theory: Part B”:

“… In 1975 … I had essentially quit my academic job at Rutgers (so I could do my research full time), and my main support came from Ames Research Center (NASA) for my work on control theory. I was also starting a publishing company, Math Sci Press, writing books for it to hold out the hope that, some day, I would get off this treadmill of endless grant proposals. (Unfortunately, it is still [March 1979] at best bearly breaking even.) …

Ever since I lost my ONR grant in 1970, thanks to Senator Mansfield, I have been trying to persuade NSF … that my work on the differential geometric foundations of engineering and physics is worthy of their support … I see my colleagues who stay within the disciplinary “clubs” receiving support much more readily … Thanks to Freedom of Information, I finally see what the great minds of my peers object to, and I see nothing but vague hearsay, bitchiness, and plain incompetence in reviewing … specialized cosed shops that blatantly discriminate against the sort of … work that I do.

I can deduce the sort of bureaucratic-administrative pressures that encourage this … However, when I look in the Physical Review today, … subjects which people … so enthusiastically supported ten years ago … [such as] “S-matrix theory” … are now dead as the Phlogiston theory …”.

About S-matrix theory, Hermann said “… the … “S-matrix theory” boom (and bust) of the 1960’s … constituted a very ingenious and compelling set of ideas whichinvolved areas of mathematics (e.g., the theory of functions of several complex variables) which were unfamiliar to physicists. They were, so to speak, led into this swamp because they were familiar with functions of one complex variable, but were naive enough not to appreciate the completely different order of complication involved in the generalization, while also not willing to invest the effort in developing and understanding the mathematics needed to make the ideas fruitful. …”.

One mathematician of those times did try to apply the mathematics of several complex variables to physics, and succeeded in getting a realistic calculation of the fine structure constant. However, the American physics community not only made no serious effort to understand and develop Armand Wyler’s work, but acts to suppress such work.

My view is that the question of “mass gap” has became a kind of legend since the proposal of Clay Institute for the prize. So, most of the work people is carrying on in the field of QCD and Yang-Mills in the infrared is blatantly ignored while solutions at this problem are pretended to come from I do not know what kind of exotic theory. An example over all is given by AdS/CFT that is often claimed to be the principal way toward answering this problem. AdS/CFT to work needs the mass gap as an input and does not prove anything about Y-M presently! This input value is generally taken from lattice computations that do not seem so reliable for pure Y-M spectrum. Time will say but I think that the true challenge implied into infrared QCD has not been completely caught by the community.

Marco

Marco,

Witten didn’t mention AdS/CFT at all, there is no mass gap there. He emphasized that the crucial problem is to show that pure QCD breaks conformal symmetry. The statement of the Millenium problem is mathematically precise and unambiguous.

Peter,

This is exactly what people working on QCD and Y-M are doing as knowing the spectra means something directly observable in accelerator facilities. I understand that Witten’s requirements are more stringent from a mathematical point of view but hints coming from physicists are generally sound enough for mathematicians to build upon. My view is that behind this problem may hide a paradigm change that, if it would have been emphasized enough, today more people would work on it rather than multiverse…

Marco

Regarding the mass-gap millennium problem, I take it that Terry Tomboulis’ proof of confinement discussed here some time ago didn’t cut it? Put another way, is he considered a candidate for the prize?

Sumar,

The rules about the Millennium prize require first of all publication of a referreed proof. As far as I can tell, Tomboulis’s paper has not been published. It also doesn’t appear to meet the usual mathematician’s standards for a complete, rigorous proof. Finally, what he is trying to show (confinement) is significantly different from what the official problem asks for, a proof that a construction of quantum YM exists satisfying a list of properties including a mass gap.

In his talk, Witten did not mention the Tomboulis paper.

Stated in another way, is it Witten who knows all research that is going around on this matter? Either, can he only know about mathematical proofs? Or people working on QCD and YM is not worth considering, just they are not mathematicians? This situation is not so good.

Marco

Marco,

Witten could have talked on many topics, but he chose this one, and made it clear that he regards understanding non-perturbative QCD as one of the central problems in theoretical physics. While he’d like to get mathematicians interested in thinking about this, I don’t think at all that he considers it a problem of purely mathematical interest.

http://de.youtube.com/watch?v=j50ZssEojtM

rowland,

Thanks for that link. I’m rofl.

Deane,

For an interesting insight into Witten’s own journey into the intimidating jungle, with Singer and Atyah guiding his steps, see Witten’s 1999 acount in “MICHAEL ATIYAH AND THE PHYSICS/GEOMETRY INTERFACE” at:

http://www.intlpress.com/AJM/p/1999/3_1/AJM-3-1-lxi-lxiv.pdf

I was delighted to read the comments about Robert Hermann since he was on my PhD committee. I read his Lie Groups for Physicists in 1975 and eventually I read all of his published books, except the self published work. His “Yang-Mills, Kaluza Klein and the Einstein Program” was especially important to me. My dissertation, “The Geometry of elementary Particles” was very much in the Hermann tradition.

When I attended Singer’s course on geometry and gauge fields at Berkley, everyone spell his nickname as “Iz”

Doubting Thomas,

So you wish me to make the point that it all depends on what your definition of ‘Is’, is?

Hmm. I think In Hollywood, setting up an obvious line like that is called ‘laying pipe’ within the industry.

At worst, Is has tricked me (and now Peter) into using the same mis-spelling of his name that he has also conned Witten, Atiyah and others into using.

I was simply trying to do the smallest of favors and have very little interest in the topic …. shall we agree not to distract ourselves further?

Best,

Eric

I didn’t mean to continue the discussion, but I was looking for the book with Witten’s article and found it at

http://www.intlpress.com/books/FoundersIndexTheory.php

And there in the table of contents I found:

Letter to Fritz, Is and Michael – R. Bott

Letter to Michael, Raoul and Iz – F Hirzebruch

But let’s leave it “AS IS”

Does anyone know if Robert Hermann is still alive and well?

He sent me an email in October 1999 saying he had been trying to understand renormalization for many years, and asking if anyone had used Colombeau theory to understand the products of operator-valued distributions that show up in quantum field theory.

In 2005 I sent him an email asking if he knew about an English translation of Felix Klein’s

Erlangen Programspeech. He never replied.(According to Thomas Love, Hermann had mentioned such a translation in a book of his. By now it’s available on the arXiv.)

Hey John,

He is alive in your old stomping ground. I will be happy to give you (or any other colleagues interested) his contact information if you like.

In his words, he’s sort of just a ‘senior’ who’s ‘given up’. He has neither email nor academic connection any more. From talking to him, I wonder whether anyone at all has seriously interviewed him about the history of the geometric physics revolution, though I could be wrong about that. There are now many more readable books on these topics and I can guess his ultimate reward.

For an objective field, it is just really really interesting how we go about subjectively creating our system of selective pressures.

Anyway, I read him a few of the comments in this thread. Thanks to all of you who had a kind word for a greying rebel. I hope I conveyed your sentiments as accurately as possible.

Regards,

Eric

The book John Baez mentioned is

Sophus Lie’s 1880 Transformation Paper (Lie Groups: History, Frontiers & Applications Series No.1) (Paperback)

Translated by M. Ackerman, with commentary by Robert Hermann

(ISBN 0915692104) Math-Sci Press