If one tried to pick a single most talented and influential figure of the past 100 years in each of the fields of pure mathematics and of theoretical physics, I’d argue that you should pick Alexander Grothendieck in pure math and Edward Witten in theoretical physics. This afternoon I’ve run across two excellent sources of information about each of them.
This week’s New Yorker has The Mysterious Disappearance of a Revolutionary Mathematician by Rivka Galchen. It’s a very well-done survey of Grothendieck’s life and work, aimed at a popular audience. If, like many mathematicians, you’ve always been fascinated by Grothendieck’s story, you won’t find too much in the article you haven’t seen before. But, if you’ve never delved into this story, you should read the article. On a related note, a copy of Récoltes et semailles that I ordered recently has just arrived in the mail, and I’m looking forward to spending some time with that this summer.
In theoretical physics a very different but equally off-the-scale talented and influential figure is Edward Witten, who is the subject of a recent long and in-depth interview by David Zierler as part of the Oral Histories program at the Niels Bohr Library and Archives.
I first met Witten probably in 1977, when I was an undergraduate at Harvard and he was a Junior Fellow, recently arrived from Princeton. Over the years since then he has done a mind-blowing quantity of highly impressive work which I’ve done my best to try to follow. You can find many places where I’ve written about this here on the blog, and there’s also a lot in my book Not Even Wrong. Much of what he discussed in the interview was familiar to me, but I learned quite a bit new from his recollection of the details of how his work came about and how he thought about it. On some of the specifics of what happened many decades ago one should keep in mind that memory is imperfect. For instance, he describes a short period as a graduate student in economics at the University of Michigan, which surprised me since in research for my book I’d read that this was at the University of Wisconsin. Maybe I got this wrong, but if so I’m not the only one (see for instance here).
Witten’s work in the area where pure mathematics and quantum field theory overlap has had an overwhelming influence on those like myself who are fascinated by both subjects and their interaction. The landscape of this area would be completely different (and highly impoverished) without him. At the same time, his equally large influence in the area of attempts to unify physics I believe has been much more problematic.
I’ll quote here with a little commentary some of the passages from the interview that I found striking or where I learned something new.
About his early years:
I was very interested in astronomy when I was growing up. Well, I was not an exception; these were the days of the Space Race, so everybody was interested in astronomy. I was given a small telescope when I was about nine or ten. That’s certainly a vivid memory. Another vivid memory is learning calculus when I was eleven. My father sort of taught me calculus or gave me materials from which I could learn it. But I didn’t advance very much in math beyond that for quite a few years…
And then [after college at Brandeis] initially you thought you would go and become an economist?
What were your interests there? Did you think that your mathematical abilities would be applied well in that field?
It’s again hard to remember reliably, but I might have thought that. And I might have also thought that I could make a contribution to international development. But I realized- well, I came to the same realization I had come to when I was working on the McGovern campaign, that it wasn’t a good match for me. I remember being very embarrassed when I told the people in the department at Michigan who had been quite kind to me, that I had decided to leave. But in hindsight, I understand something that wasn’t that clear to me at the time, that if a given graduate program isn’t a good match for a given student, the department and the student are both better off if that’s realized sooner rather than later. If I had understood that at the time, I would have been less embarrassed, probably, with what I told them.
How to learn general relativity in ten days:
Was general relativity considered popular or interesting at Princeton at the time that you were a graduate student?
Well, I was certainly interested in it. I learned general relativity in a very exciting period of about ten days, from the book of Steve Weinberg. I mean, I tried to learn more from the book of Misner, Thorne, and Wheeler, and I did learn more from it, but my opinion of the book was what it remains now, which is that it’s got a lot of great stuff in it, but it’s a little bit hard to use it to learn systematically. The book I found useful for studying systematically was by Steve Weinberg.
The Harvard Society of Fellows:
Ed, did you enjoy the Harvard Society of Fellows, the social aspects of it?
Well, I enjoyed it up to a point, but let’s just say that many other people thrive on that more than I did.
On how he experienced the First Superstring Revolution.
I’m not exactly sure what I would have said if you had asked me. There’s no interview, so there’s no record of my thinking in 1982 or 1983, and I won’t be able to remember very well. But as I was telling you, I was interested enough to spend a whole summer reading John Schwarz’s review article, but a little bit wary of becoming too involved in it…
Something that was obvious to me but wasn’t immediately completely obvious to everybody was this. Green and Schwarz had put string theory in the form where there was a very strong case that there was a consistent quantum theory that described gravity together with other forces. And the other forces could be gauge fields, somewhat like in the Standard Model. But there was something extremely conspicuous that was wrong in terms of phenomenology, and that was that the weak interactions couldn’t violate parity…
And as it existed in 1982 and 1983, string theory was a consistent theory of gravity unified with other forces, but it completely missed the chiral structure. So, to me, that was a huge siren blaring. Anyway, to set the stage, I want to just point out to you that it was clear by 1982 or 1983 that there were an incredible variety of delicate things that fit together perfectly to make it possible to have a theory of quantum gravity based on string theory. It was unbelievable that it could all be a coincidence. Yet it was markedly wrong for describing the real world because of this question of the chiral nature of the fermions. But then in 1984, Green and Schwarz discovered a more general method of anomaly cancellation, and everything changed…
So, anyway, what was really problematical for Green and Schwarz was the combination of fermion chirality and anomalies. Taking these together, it seemed that string theory could not work. But then, in August 1984, Green and Schwarz discovered a new mechanism for anomaly cancellation, and everything changed…
So, it was immediately obvious to me, once they made their discovery, that you could make at least semi-realistic models of particle physics, in that framework. But also, to me, I had done kind of an experiment in the following sense. I had spent two years watching this, wondering, could it be? Can it be that all the coincidences that had been discovered that made string theory possible were just coincidences? As far as I was concerned, the discovery they made in 1984 was an empirical answer of “no” to that question. If the miraculous-looking things that had been discovered up to 1982 and 1983 were truly coincidences, you’d then predict there wouldn’t be any more such coincidences. That had proved to be wrong when they made this miraculous-looking discovery about anomalies that enabled the theory to be much more realistic.
In explaining this to you, I’m trying to help you understand why this had so much of an impact on my thinking, watching from the outside for a couple years, wondering if this subject was as amazing as it appeared to me. And a “no” answer would have predicted there shouldn’t be more miraculous discoveries. And that was, to my satisfaction, disproved in August 1984. So, after that, the hesitation that had kept me from becoming more heavily involved earlier evaporated. Now, I realized that in the physics world, there were plenty of people who hadn’t lived through this two years of uncertainty that I had lived through, and in many cases they had never heard of the whole thing until August 1984. And they hadn’t done the experiment I had done. So, they didn’t react as I did.
Here Witten explains how one very specific technical calculation triggered for him a dramatic vision of a possibility of a unified theory of everything, a vision that has stayed with him to this day, nearly four decades later.
About his evangelism for string theory unification starting in 1984:
How much cheerleading did you do among your colleagues, both near and far, after this revolution in 1984, that this is what people should concentrate on? That we can have this figured out in the near term?
I wasn’t intentionally cheerleading, but I was very enthusiastic. And I actually think I was right to be enthusiastic. I wasn’t intentionally cheerleading, but to the extent that I encouraged other people to get involved, I’ve got no regrets about it at all (laughter).
Another very interesting recent interview in the same series is one with Cumrun Vafa. Here’s what Vafa remembers about that time:
I remember I was at my office, I had come back from a trip, from I think the summer school in Europe, in Italy. Had come back to my office in Princeton on the fourth floor, and Ed’s office is on the third floor. And he rarely came to our floor, fourth floor, but here he was, coming and knocking at my door, and then saying, “Have you heard about the revolution?”…
I said, “What revolution?” He said, “The SO(32) revolution.” Okay, that was my first introduction to Green and Schwarz’s work. SO(32) revolution. I said, “No, what is it?” He said, and he was completely sure, confident, that physics is not going to be the same after this. He said, “Physics is going to change forever because of this, and now everybody is going to work on this.”
I had left Princeton for Stony Brook early that summer. During the next few years, reports I got from fellow postdocs who tried to talk to Witten about their work were pretty uniformly something like “he told me that what I’m doing is all well and good, but that I really should be working on string theory.”
Unlike the case in the interview with Vafa previously mentioned, Zierler doesn’t really try and pin Witten down on the subject of the problems of string theory. He does ask:
What was happening at the time or has happened since in the world of experimentation or observation that may get us closer to string theory being testable?
but lets Witten give a non-answer, which in effect is that the landscape means string theory unification is completely untestable, so he has pretty much given up:
So, if you talked to me in the 1980s, I’m sure I would have expressed some hope about seeing supersymmetry as part of the answer of the hierarchy problem. But I would have expressed a lot of confidence about observing something that would have explained the hierarchy problem. …
But ultimately, with the LHC, experiment has reached the point that it’s extremely problematic to have what’s called a natural explanation of the weak scale, a mechanism that would explain in a technically natural way why the Higgs particle is as light as it is, thus making all the particles light. It’s actually a baby version of the problem with the cosmological constant. So, to the extent that the multiverse is a conceivable interpretation of why the universe accelerates so slowly, it’s also a conceivable explanation of why the weak scale is so small. It might be the right interpretation. But if it is, it’s not very encouraging for understanding the universe. When the multiverse idea became popular around 1999, 2000, and so on, I was actually extremely upset, because of the feeling that it would make the universe harder to understand. I eventually made my peace with it, accepting the fact that the universe wasn’t created for our convenience.
You would think that having an untestable theory on your hands would mean that you would try something else, anything else, but Witten seems convinced that whatever its problems, it’s the only way forward:
[About the second superstring revolution and M-theory] It’s satisfying to know that there was only one candidate for superunification. There’s only one reasonable candidate now for the theory that combines gravity and quantum mechanics. Before 1995, there was more than one. It’s more satisfying to know that the theory seems to have a lot of possible manifestations, in terms of approximate vacuum states, but at a fundamental level, there’s only one fundamental theory or system of equations, that we admittedly don’t understand very well. That’s got to be an advance of some kind…
By the time he [Einstein] had the theory [GR], he had the right mathematical framework of Riemannian geometry. At least by the time the theory was invented, he had the ideas it was based on, and some of them he had had before.
String theory and M-theory have always been different. From the beginning, they were discovered by people who discovered formulas or bits and pieces of the theory without understanding what’s behind it at a more fundamental level. And what we understand now, even today, is extremely fragmentary, and I’m sure very superficial compared to what the real theory is. That’s the problem with the claim that supposedly I invented M-theory. It would make at least as much sense to say that M-theory hasn’t been invented yet. And you could also claim it had been invented before by other people. Either of those two claims is defensible (laughter). So I made some incremental advances in a subject that’s far from being properly understood.
This “we don’t know what the fundamental equations are, but we know that they are unique” argument has never made any sense to me.
On his relation to mathematics:
What did it feel like to win the Fields Medal as a physicist?
Well, it was a thrill, of course. It felt a little funny because I knew that obviously I was a non-standard selection. And I don’t like controversy about science, and I felt that I might have been a controversial choice in the math world. But on the other hand, I hadn’t selected myself, so I didn’t feel any controversy was my fault…
What’s a little funny about my relation to the math world is that although some of my papers are of mathematical interest, they rarely have the detail of math papers. And I can’t provide that detail. I simply don’t have the right background. What I bring to the subject is an ability to understand what quantum field theory or string theory have to say about a math question. But quantum field theory and string theory are not in the precise mathematical form where such statements can usually be rigorous.
The “I don’t like controversy about science” quote makes clear that Witten and I are temperamentally very different…
About the birth of geometric Langlands:
By the late 1980s- I’m probably forgetting bits of the story, I should tell you- but by the late 1980s, Sasha Beilinson and Vladimir Drinfeld had discovered what they called a geometric version of the Langlands program, and it involved ingredients of quantum field theory. Tantalizing. But it was tantalizing because they were using familiar ingredients of quantum field theory in a very unfamiliar way. It looked to me as if somebody had put the pieces at random on a chess board. The pieces were familiar, but the position didn’t look like it could happen in a real chess game. It just looked crazy. But anyway, it was clear it had to mean something in terms of physics. I even worked on that for a while at the time.
I think I’ve gotten this slightly out of order. I think when I worked on it was actually before the work of Beilinson and Drinfeld, driven by other clues. And the Beilinson and Drinfeld work was one of the things that made me stop, because I realized that A, I couldn’t understand what they were doing at the time, and B, there were too many things I didn’t know that they knew, and that seemed to be part of the story. Anyway, as you can see, my memories from whatever happened in the late 1980s are pretty scrambled.
They wrote a famous paper that was never finished and never published. It’s 500 pages long. You can find it online, if you like. They have an incredibly generous acknowledgement of what they supposedly learned from me, which is way exaggerated. Based on a hunch, I told them about a paper of Nigel Hitchin, but I didn’t understand anything of what they attributed to me. At any rate, regardless, even if I didn’t understand what they did with it, the fact that I was able to point them to the right paper was another sign of the fact that what they were doing had something to do with the physics I knew. But I couldn’t make sense of the connection. And this kept nagging at me off and on for a long time.
He then goes on to tell the story of the IAS workshop on geometric Langlands and how it led to his work on a QFT version of geometric Langlands.
In recent years Witten has continued to work on geometric Langlands and other topological quantum field theory related topics at the mathematics end of things. As far as physics goes, he is following the very popular “it from qubit”, quantum gravity from information theory, line of thinking:
And the third time [revolutions: first and second were two superstring revolutions] has been the last six or seven years. It’s actually hard to remember the evolution of my thinking (laughter). I reread an interview I had done in 2014 which told me what my thinking was in 2014 better than I could have remembered it reliably (laughter). And what I told the interviewer at that time was somewhat similar to what I’m telling you right now. So, this has gone on for a while, and despite that, I haven’t really found the right way to become involved myself. But I do suspect that something big is happening.
What has happened since 2014, when you initially got excited about this?
There have been various striking developments, but a particularly dramatic one came in 2019 when there was success in understanding what is known as the Page curve in black hole evaporation… Lots of things have happened that show that there’s a conspiracy between gravity and quantum mechanics. Somehow gravity at the classical level knows about quantum mechanics and statistical mechanics…
To bring the conversation right up to the present, as we discussed right at the beginning, your interest in quantum information. And you said you don’t yet know how you might break into the field. What might be some possible avenues?
Well, when I was a graduate student, I sat down one day with piles of paper preprints. We didn’t have the archive. I’d sit down with piles and piles of paper preprints, and go through them, trying to find something I might do. The most interesting calculation I did as a student- I told you about it- was this calculation of deep inelastic photon-photon scattering, which was inspired by a paper I saw by Roger Kingsley, who studied the question but not quite with the most modern QCD ideas. So, when I was a graduate student trying to break in, I would go through piles of preprints. I guess the equivalent now is to look at papers in the archive and try to see what I might do. And I have made some minor contributions, actually, but I don’t feel like I’ve fully become engaged with the subject, as I have with other subjects in the past.
Witten and the interviewer discuss the difficulty of finding something to work on that is not too hard but still significant, and he comments that this is:
…the difficulty I’ve had getting involved with quantum information theory and gravity. I found a few things that I could do, but they were a little bit too narrow to really make me think that I was getting involved where I wanted to. And I haven’t quite found the right avenue. But I haven’t given up (laughter). I do have the feeling that’s the direction where something big is most likely to happen. You see, there isn’t a general understanding of what string/M-theory mean. And there’s something missing in the general understanding of quantum gravity. The biggest hope would be that those two would somehow make contact with each other.
I can understand why Witten hopes that the mystery of quantum information theory and gravity will give insight into and resolve the mystery of what M-theory is, finally vindicating his 1984 vision, but this looks to me like a very, very long shot.