Yet More Geometric Langlands News

It has only been a couple weeks since my last posting on this topic, but there’s quite a bit of new news on the geometric Langlands front.

One of the great goals of the subject has always been to bring together the arithmetic Langlands conjectures of number theory with the geometric Langlands conjectures, which involved curves over function fields or over the complex numbers. Fargues and Scholze for quite a few years now have been working on a project that realizes this vision, relating the arithmetic local Langlands conjecture to geometric Langlands on the Fargues-Fontaine curve. Their joint paper on the subject has just appeared [arXiv version here]. It weighs in at 348 pages and absorbing its ideas should keep many mathematicians busy for quite a while. There’s an extensive introduction outlining the ideas used in the paper, including a long historical section (chapter I.11) explaining the story of how these ideas came about and how the authors overcame various difficulties in trying to realize them as rigorous mathematics.

In other geometric Langlands news, this weekend there’s an ongoing conference in Korea, videos here and here. The main topic of the conference is ongoing work by Ben-Zvi, Sakellaridis and Venkatesh, which brings together automorphic forms, Hamiltonian spaces (i.e classical phase spaces with a G-action), relative Langlands duality, QFT versions of geometric Langlands, and much more. One can find many talks by the three of them about this over the last year or so, but no paper yet (will it be more or less than 348 pages?). There is a fairly detailed write up by Sakellaridis here, from a talk he gave recently at MIT.

In Austin, Ben-Zvi is giving a course which provides background for this work, bringing number theory and quantum theory together, conceptualizing automorphic forms as quantum mechanics on arithmetic locally symmetric spaces. Luckily for all of the rest of us, he and the students seem to have survived nearly freezing to death and are now back at work, with notes from the course via Arun Debray.

For something much easier to follow, there’s a wonderful essay on non-fundamental physics at Nautilus, The Joy of Condensed Matter. No obvious relation to geometric Langlands, but who knows?

Update: Arun Debray reports that there is a second set of notes for the Ben-Zvi course being produced, by Jackson Van Dyke, see here.

Update: David Ben-Zvi in the comments points out that a better place for many to learn about his recent work with Sakellaridis and Venkatesh is his MSRI lectures from last year: see here and here, notes from Jackson Van Dyke here.

Posted in Langlands | 9 Comments

Isadore Singer 1924-2021

I was sorry to hear this morning of the death yesterday at the age of 96 of Is Singer, a mathematician who led much of the interaction between mathematics and physics during the 1970s and 1980s. In the early stages of my career, among mathematicians investigating the amazing relations between mathematics and the quantum field theories describing fundamental physics there were three towering figures: Atiyah, Bott and Singer. That the last of them has now left us marks the end of an era.

Each of the three had a huge influence on me, both intellectually and personally. Reading their papers and listening to their lectures were great intellectual experiences, shaping early on my understanding of what is central to mathematics and how it fits together with physics. Especially inspirational was the way that they brought together very different fields of mathematics, with Atiyah having his roots in algebraic geometry, Bott in topology and Singer in analysis. Their work together makes a strong case for the unity of mathematics and the relation to physics makes an equally strong case for the unity of mathematics and physics.

On a personal level, at a time when I was tentatively moving from a career in physics to one in mathematics, getting to meet and talk to each of them had a big impact. Much as I respected the great theoretical physicists I had met, rarely had I found them to be particularly friendly or encouraging, and their attitudes influenced the general atmosphere of the field. Atiyah, Bott and Singer struck me each in their own way as wonderfully warm and enthusiastic personalities, and I believe this influenced the atmosphere among mathematicians working in their fields. They were among the most respected figures in the math community, so their enthusiasm for ideas coming out of physics generated a lot of interest in these ideas among a wide variety of mathematicians.

Singer had always had an interest in physics, majoring in physics as an undergraduate at the University of Michigan, then after the war going to graduate school in mathematics at the University of Chicago. I highly recommend reading or watching this long interview with him from 2010, where you can learn the story of his career.

A mathematical high point of this career was his work during the early 1960s with Atiyah that led to the Atiyah-Singer index theorem. A crucial part of this story was Atiyah in 1962 asking Singer why the A-roof genus was integral. Singer realized that this was because it counts the number of solutions of an equation, and that the equation was the Dirac equation. This example in some sense generates a huge amount of mathematics which is described by the index theorem, and which links together very different mathematical fields. On this and other topics, well-worth reading is the 2004 interview with Atiyah and Singer after they were awarded one of the first Abel Prizes.

One can trace much of the history of the modern interaction of mathematics and quantum field theory to an origin back in the summer of 1976, when Singer visited Stony Brook and talked to physicists there about gauge theories, geometry and the BPST instanton (Simons and Yang a year earlier had started to realize how gauge theory, geometry and topology were linked). The next year he was in Oxford working with Atiyah and Hitchin on instantons, which really set off an explosive development of new ideas, inspiring and fascinating both mathematicians and physicists.

Singer spent the years from 1977 to 1983 at Berkeley, which he turned into a major center for this new mathematical physics. During this time he was also one of the founders of MSRI, which to this day plays a major role in worldwide mathematical research. After 1983 he returned to MIT, from which he retired in 2010. I believe the last time I saw him was at his 85th birthday conference, which I wrote about here.

Update: The New York Times has an obituary here.

Posted in Obituaries | 9 Comments

Geometric Langlands News

There’s various news to report on the geometric Langlands front, spanning number theory to quantum field theory:

Minhyong Kim has been running an Online Mini-Conference on the Geometric Langlands Correspondence for the past month, and Dennis Gaitsgory has been doing something similar since last spring at his Geometric Langlands Office Hours.

Very recently Edward Frenkel has given talks in both places (see talks here, here and here, slides here and here). He’s been talking about joint work with Etingof and Kazhdan on a function-theoretic (as opposed to sheaf-theoretic) version of geometric Langlands. They have a paper out here, are working on two more.

This work to some extent has its origins in attempts by Langlands to come up with his own version of such a function-theoretic approach. Frenkel was asked to discuss this topic by the organizers of the Abel Conference in honor of Langlands. I wrote about what happened here. Frenkel came to the conclusion that what Langlands was suggesting could not work (Langlands vehemently disagreed…), but this led him to the current research he is pursuing with Etingof and Kazhdan. For a written version of Frenkel’s talk explaining all this, see here.

On the quantum field theory front, Witten and Gaiotto have been working on relating older ideas of Gukov-Witten about using branes as a general method of quantization, applying this to geometric Langlands, in the new context that Frenkel’s talks discuss. Witten talked about this last week in the Kim seminar (video here, slides here). Gaiotto last week also spoke about this at a Kansas State seminar, video here, slides here.

The original 2008 Gukov-Witten paper on branes and quantization is here, Gukov’s 2010 Takagi lectures on this are written up here. The problem of how to quantize a general symplectic manifold is a fascinating one, and at the time I was very interested to see this proposal. It does however invoke a very sophisticated set of ideas about quantum field theories in order to deal with what one would think are much simpler examples of the quantization problem. Perhaps this program would come into its own in this new case, where the quantization problem involves similarly sophisticated mathematical constructions.

From another side of the geometric Langlands world, Peter Scholze is continuing his lectures on his ongoing work with Laurent Fargues that reformulates the local Langlands correspondence in terms of geometric Langlands on the Fargues-Fontaine curve. There are associated discussion sections, with a web-page here.

Announcement: I’d been reading about how the hot new idea for authors on the internet is Substack, where all sorts of interesting material can now be found. After thinking about this “back to the email newsletter” model for a minute, I realized that I should try and see if I could get email subscriptions to this blog working. There’s now a place over on the right where you can ask for an email subscription. No experience with this yet, so I can’t guarantee either that it works or that problems won’t turn up that will cause me to have to turn that feature off.

Update: For another talk by Witten about this from today (Feb. 11) see here.

Posted in Langlands | 7 Comments

This Week’s Hype

I had just been thinking the other day about how little one hears recently about the multiverse, with those previously involved in heavy promotion of the idea perhaps having thought better of it. Today however, Quanta has Physicists Study How Universes Might Bubble Up and Collide. This describes work of a sort that has become popular in recent years: study of various condensed matter systems, with a huge dollop of hype on top about quantum gravity based on some aspect of the condensed matter theory calculation having some vague relation to some calculation in some toy quantum gravity model or other.

I’ve written extensively here and elsewhere about the real problem with all claims by theorists to be studying the multiverse: they’re Theorists Without a Theory, lacking any sort of viable theory which could make the usual sort of scientific predictions. The main problem with the Quanta article is at the beginning:

What lies beyond all we can see? The question may seem unanswerable. Nevertheless, some cosmologists have a response: Our universe is a swelling bubble. Outside it, more bubble universes exist, all immersed in an eternally expanding and energized sea — the multiverse.

The idea is polarizing. Some physicists embrace the multiverse to explain why our bubble looks so special (only certain bubbles can host life), while others reject the theory for making no testable predictions (since it predicts all conceivable universes). But some researchers expect that they just haven’t been clever enough to work out the precise consequences of the theory yet.

Now, various teams are developing new ways to infer exactly how the multiverse bubbles and what happens when those bubble universes collide.

The big problem is with:

they just haven’t been clever enough to work out the precise consequences of the theory yet.

The reference to “precise consequences” is a common misleading rhetorical move, implying that there is no problem getting “imprecise consequences”, that the problem is just getting those extra digits of numerical precision. What’s really going on is that we know of no theoretical consequences of the multiverse, precise or imprecise, because there is no viable theory. The logic here is pretty much pure wishful thinking: if you look at colliding Bose-Einstein condensates and see a particular pattern, then if you saw a pattern like that in the CMB, you could try and infer something about your unknown multiverse theory. It’s not unusual for theorists to work on speculative ideas involving some degree of wishful thinking, but this is a case of taking that to an extreme.

Update: One of the very few theorists who has pushed back on the multiverse ideology is Paul Steinhardt. Howard Burton has posted here something from his interviews with Steinhardt, which includes this from Steinhardt:

“I’ve had this discussion where I’ll say, ‘Well, what do you think about the multiverse problem?’ and they reply, ‘I don’t think about it.’

“So I’ll say, ‘Well, how can you not think about it? You’re doing all these calculations and you’re saying there’s some prediction of an inflationary model, but your model produces a multiverse — so it doesn’t, in fact, produce the prediction you said: it actually produces that one, together with an infinite number of other possibilities, and you can’t tell me which one’s more probable.’

“And they’ll just reply, ‘Well, I don’t like to think about the multiverse. I don’t believe it’s true.’

“So I’ll say, ‘Well, what do you mean, exactly? Which part of it don’t you believe is true? Because the inputs, the calculations you’re using — those of general relativity, quantum mechanics and quantum field theory — are the very same things you’re using to get the part of the story you wanted, so you’re going to have to explain to me how, suddenly, other implications of that very same physics can be excluded. Are you changing general relativity? No. Are you changing quantum mechanics? No. Are you changing quantum field theory? No. So why do you have a right to say that you’d just exclude thinking about it?’

“But that’s what happens, unfortunately. There’s a real sense of denial going on.”

Update: Ethan Siegel has an excellent piece on the basic problem with string theory (to the extent it’s well-defined, it has too large a (super)symmetry group and too many dimensions, no explanation for how to recover 4 space-time dimensions and observed symmetry groups).

Here’s why the hope of String Theory, when you get right down to it, is nothing more than a broken box of dreams.

Update: If you’re looking for a detailed discussion of multiverse theories, of neither the usual promotional sort, nor the highly critical sort I specialize in, I can recommend Simon Friedrich’s new book Multiverse Theories: A Philosophical Perspective. Friedrich has a blog entry about the book here.

Posted in This Week's Hype | 22 Comments

What is a Spinor?

Recently Jean-Pierre Bourguignon recently gave the Inaugural Atiyah Lecture, with the title What is a Spinor? The title was a reference to a 2013 talk by Atiyah at the IHES with the same title. Bourguignon’s lecture is not yet online, but I realized there are lectures explaining what a spinor is that I can highly recommend: my own, in this semester’s course on the mathemematics of quantum mechanics. I’m closely following the textbook I wrote.

Teaching this course this past academic year has made me all too aware of things that are less than ideal about the book, and I unfortunately haven’t had time to get to work on making any significant improvements. Going through the material on spinors though, I’m pretty happy with how that part of the book turned out, think it provides a clear explanation of a beautiful and important story, one that is not readily available elsewhere.

One aspect of this that I emphasize is the remarkable parallel between

  • The usual story of canonical quantization, which is based on an antisymmetric bilinear form on phase space, giving an algebra of operators generated by $Q_j,P_j$ acting on the usual quantum state space.
  • Replacing antisymmetric by symmetric, you get the Clifford algebra, generated by $\gamma$-matrices, acting on the spinors.

For a table summarizing precisely this parallelism, see chapter 32 of the book.

For more video from my office, I recently had a long conversation with Reza Katebi, who has a Youtube channel of interviews called The Edge of Science.

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Wilczek on the Problems of Fundamental Physics

Sean Carroll has a new interview up with Frank Wilczek in which they discuss, among other things, the problematic current state of fundamental physics. On the topic of string theory, here’s the discussion:

0:58:34.8 SC: Well, some of this worry has come out of string theory, many of our colleagues for the last several decades have pointed to string theory as the most promising way forward. As far as I know, you have not done a lot of work directly on conventional string theory. What is your feeling about that approach to moving beyond quantum field theory?

0:58:54.8 FW: Well, I think it has produced a lot of attractive work that’s intellectually rich and has spun off into fertile mathematics, but I don’t see that it’s been converging towards informative assertions about the physical world…

0:59:18.7 SC: That’s very elegantly stated, actually, yes.

0:59:21.0 FW: That you can check. And for me personally, I’ve kind of voted with my feet, I think there are more promising things to think about, that’s partially a sociological statement, but I think it’s, string theory is getting plenty of attention, it doesn’t need me. I’m happier doing things that other people aren’t doing, but that’s a personal statement, and so far, I haven’t regretted my choice, but I watch what people… I watch the subject and I watch what people are doing and I wish them good luck, and if and when things that I think are promising insights into the physical world emerge, I will pay a lot of attention.

1:00:12.3 SC: Sure Do you think that the rest of the field has voted with their feet in a slightly too uniform way, do you think that too much of our intellectual effort is going in that particular direction?

1:00:21.8 FW: I do, but I might be wrong, so I don’t want to discourage. Plenty of people are doing other things, so it’s not as if the rest of the world is feeling the lack of input from people who are working on string theory, it’s fine, people can work on string theory and it doesn’t hurt anything. I feel… Well, it’s going to sound, I don’t want to be patronizing, the people who do it are mostly adults and they know what they’re doing, but students and people who are thinking about what they’re going to do should go into it with open eyes. They should realize that the prospect of making an impact in our understanding of empirical science or technology are not… The prospect that you’ll make impact like that is probably not optimized by going into string theory.

1:01:20.8 SC: Yeah, no, actually, I think that we’re in exact alignment here. I feel a need to defend the string theory against unfair criticisms, but I do worry a little bit about the fact that it seems hard these days to connect it directly to empirical reality.

1:01:36.2 FW: Yeah, well, some nice ideas are coming off, as coming out as spin-offs, very, very clever people do string theory and they do clever things. So as I said, there’s been a lot of fruitful mathematics, there have been new techniques that have proved somewhat useful in condensed matter, although certainly not proportional to the amount of effort that’s going into it, and the future may look different, they may be real breakthroughs that come out of string theory that wouldn’t have come otherwise. But so far, the amount, I would say, other people may disagree, and I might be very unpopular among some of my colleagues for saying this, but I think the output compared to the input has been pretty disappointing on the empirical side.

I find it kind of remarkable that Carroll, known for defending string theory and string theorists, here reacts to Wilczek’s pretty negative characterization of string theory with “I think that we’re in exact alignment here.”

About current hot topic work on the black hole information paradox:

1:02:33.5 SC: Right. You have been involved in productive ways on the black hole information problem, which a lot of string theorists care about… What is your current feeling on the state of that problem? Do you think we’re making real progress?

1:02:50.8 FW: I think progress is being made in the sense that more intellectually coherent pictures are being drawn and some surprising connections to error correction and really interesting new chapters of quantum theory are emerging. On the other hand, it is a very esoteric problem, nobody’s going to produce… I don’t see a way, but who knows, but nobody has produced an experimental system to which these ideas apply in any reasonably direct way. So what does it mean to solve a problem like that? I’m not even sure what it means, where you can’t check. Many hypotheses go into it, the distance between the models and actual black holes that were phenomena you can observe are vast and many things could go wrong along the way in making these models.

1:04:04.7 FW: So I guess, yeah, it’s wonderful that people are making progress at the field, they’re making progress and have a literature that they enjoy, and it really is interesting from any point of view, it’s good, and maybe I should leave it at that, but how should I say? I don’t think it’s… I don’t think it’s the pinnacle of physics, let me put it… Let me put it that way.

On SUSY, Wilczek acknowledges

I’m a supersymmetry diehard.

which he certainly is.

If you look back at his many talks about prospects for the future, you’ll see that pre-LHC he was arguing

By ascending a tower of speculation, involving now both extended gauge symmetry and extended space-time symmetry, we seem to break though the clouds, into clarity and breathtaking vision. Is it an illusion, or reality? This question creates a most exciting situation for the Large Hadron Collider(LHC), due to begin operating at CERN in 2007, for this great accelerator will achieve the energies necessary to access the new world of of heavy particles,if it exists.

In the current interview and elsewhere, Wilczek makes clear the reason he believes in SUSY is his 1981 calculation with Dimopoulos/Raby showing that in SUSY versions of GUTs you could get the coupling constant evolution to overlap at the same energy. He’s still quite devoted to this argument, for him it’s of greater significance than the usual hierarchy problem arguments.

He has by now lost multiple bets that the LHC would see SUSY particles, including ones with Garrett Lisi in 2009 and Tord Ekelöf in 2012. At this point, even diehards like Wilczek acknowledge that chances that the LHC will see SUSY are slim. Another problem is that increasingly sensitive proton decay experiments have also ruled out a large part of the proton lifetimes predicted by the SUSY models Wilczek favors. He puts his faith in further proton decay experiments and a new, expensive collider. This is pretty much exactly the sort of thing that causes Sabine Hossenfelder to go ballistic over arguments for a new collider.

For a flavor of the SUSY discussion, here’s one piece of it, with Carroll starting off with a quite peculiar argument for SUSY:

0:29:28.8 SC: Yeah, maybe you can opine on this, but the way that I like to say it is, we could, in the space of all possible worlds that we live in, only one of them, we could have found supersymmetry already at the LHC very easily, but the fact that we haven’t doesn’t mean it’s not there. Maybe it’s less likely that it’s there, but it’s easy also to imagine theories where supersymmetry is real, and we just haven’t seen it yet.

0:29:53.1 FW: Right. So supersymmetry, as I said, there have to be… For supersymmetry to be valid, there have to be these superpartner particles that are the particles that the particles we know about turn into when they move into the quantum dimensions, but we don’t know what their masses are. We know some of their properties, but not their masses, and they could be very heavy. If they’re very, very heavy, we lose the benefit of improving… The benefit that supersymmetry would otherwise give in improving how the couplings unify, but okay, maybe that was a cruel joke on the part of nature. I want to think not, but the alternative is that they’re just a little bit too heavy to have been produced easily and identified easily at the LHC, and we just have to work a little bit harder and spend a little more money on…

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Is the Standard Model Just an Effective Field Theory?

An article by Steven Weinberg entitled On the Development of Effective Field Theory appeared on the arXiv last night. It’s based on a talk he gave in September, surveys the history of effective field theories and argues for what I’d call the “SM is just a low energy approximation” point of view on fundamental physics. I’ve always found this point of view quite problematic, and think that it’s at the root of the sad state of particle theory these days. That Weinberg gives a clear and detailed version of the argument makes this a good opportunity to look at it carefully.

A lot of Weinberg’s article is devoted to history, especially the history of the late 60s-early 70s current algebra and phenomenological Lagrangian theory of pions. We now understand this subject as a low energy effective theory for the true theory (QCD), in which the basic fields are quarks and gluons, not the pion fields of the effective theory. The effective theory is largely determined by the approximate SU(2) x SU(2) chiral flavor symmetry of QCD. It’s a non-linear sigma model, so non-renormalizable. The non-renormalizability does not make the theory useless, it just means that as you go to higher and higher energies, more possible terms in the effective Lagrangian need to be taken into account, introducing more and more undetermined parameters into the theory. Weinberg interprets this as indicating that the right way to understand the non-renormalizability problem of quantum gravity is that the GR Lagrangian is just an effective theory.

So far I’m with him, but where I part ways is his extrapolation to the idea that all QFTs, in particular the SM, are just effective field theories:

The Standard Model, we now see – we being, let me say, me and a lot of other people – as a low-energy approximation to a fundamental theory about which we know very little. And low energy means energies much less than some extremely high energy scale 1015−1018 GeV.

Weinberg goes on to give an interesting discussion of his general view of QFT, which evolved during the pre-SM period of the 1960s, when the conventional wisdom was that QFTs could not be fundamental theories (since they did not seem capable of describing strong interactions).

I was a student in one of Weinberg’s graduate classes at Harvard on gauge theory (roughly, volume II of his three-volume textbook). For me though, the most formative experience of my student years was working on lattice gauge theory calculations. On the lattice one fixes the theory at the lattice cut-off scale, and what is difficult is extrapolating to large distance behavior. The large distance behavior is completely insensitive to putting in more terms in the cut-off scale Lagrangian. This is the exact opposite of the non-renormalizable theory problem: as you go to short distances you don’t get more terms and more parameters, instead all but one term gets killed off. Because of this, pure QCD actually has no free parameters: there’s only one, and its choice depends on your choice of distance units (Sidney Coleman liked to call this dimensional transvestitism).

The deep lesson I came out of graduate school with is that the asymptotically free part of the SM (yes, the Higgs sector and the U(1) are a different issue) is exactly what you want a fundamental theory to look like at short distances. I’ve thus never been able to understand the argument that Weinberg makes that at short distances a fundamental theory should be something very different. An additional big problem with Weinberg’s argument is its practical implications: with no experiments at these short distances, if you throw away the class of theories that you know work at those distances you have nothing to go on. Now fundamental physics is all just a big unresolvable mystery. The “SM is just a low-energy approximation” point of view fit very well with string theory unification, but we’re now living with how that turned out: a pseudo-scientific ideology that short distance physics is unknowable, random and anthropically determined.

In Weinberg’s article he does give arguments for why the “SM just a low-energy approximation” point of view makes predictions and can be checked. They are:

  • There should be baryon number violating terms of order $(E/M)^2$. The problem with this of course is that no one has ever observed baryon number violation.
  • There should be lepton number violating terms of order $E/M$, “and they apparently have been discovered, in the form of neutrino masses.” The problem with this is that it’s not really true. One can easily get neutrino masses by extending the SM to include right-handed neutrinos and Dirac masses, no lepton number violation. You only get non-renormalizable terms and lepton number violation when you try to get masses using just left-handed neutrinos.

He does acknowledge that there’s a problem with the “SM just a low-energy approximation to a theory with energy scale M=1015−1018 GeV” point of view: it implies the well-known “naturalness” or “fine-tuning” problems. The cosmological constant and Higgs mass scale should be up at the energy scale M, not the values we observe. This is why people are upset at the failure of “naturalness”: it indicates the failure not just of specific models, but of the point of view that Weinberg is advocating, which has now dominated the subject for decades.

As a parenthetical remark, I’ve today seen news stories here and here about the failure to find supersymmetry at the LHC. At least one influential theorist still thinks SUSY is our best hope:

Arkani-Hamed views split supersymmetry as the most promising theory given current data.

Most theorists though think split supersymmetry is unpromising since it doesn’t solve the problem created by the point of view Weinberg advocates. For instance:

“My number-one priority is to solve the Higgs problem, and I don’t see that split supersymmetry solves that problem,” Peskin says.

On the issue of quantum gravity, my formative years left me with a different interpretation of the story Weinberg tells about the non-renormalizable effective low-energy theory of pions. This got solved not by giving up on QFT, but by finding a QFT valid at arbitrarily short distances, based on different fundamental variables and different short distance dynamics. By analogy, one needs a standard QFT to quantize gravity, just with different fundamental variables and different short distance dynamics. Yes, I know that no one has yet figured out a convincing way to do this, but that doesn’t imply it can’t be done.

Update: I just noticed that Cliff Burgess’s new book Introduction to Effective Field Theory is available online at Cambridge University Press. Chapter 9 gives a more detailed version of the same kind of arguments that Weinberg is making, as well as explaining how the the Higgs and CC are in conflict with the effective field theory view. His overall evaluation of the case
“Much about the model carries the whiff of a low energy limit” isn’t very compelling when you start comparing this smell to that of the proposals (SUSY/string theory) for what the SM is supposed to be a low energy limit of.

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Various Links

Our semester at Columbia started earlier than usual this year, with first classes this week, my first class yesterday. This semester I’m teaching the second half of a year-long course on the mathematics of quantum mechanics. There’s a Youtube channel with the lectures for the first half of the course, and now also for the second half. The course is largely following the textbook I wrote based on teaching this is earlier years. The first lecture yesterday was a summary of a point of view on canonical quantization explained in the first semester and in the book. This point of view is essentially that Hamiltonian mechanics is based on a Lie algebra (functions on phase space with Poisson bracket the Lie bracket), and canonical quantization is all about the essentially unique unitary representation of (a subalgebra of) that Lie algebra. On Thursday I’ll start on the fermionic version of canonical quantization, which has a very much parallel structure, giving a super-Lie algebra and spinors.

A few other items:

  • John Baez’s This Week’s Finds in Mathematical Physics was an unprecedented project conducted over 17 years, providing a wealth of fantastic expository material on topics in math and physics. It started in 1993, and on its twentieth anniversary I wrote an appreciation (in an appropriate font) here. John has now announced that this material has been typeset (2610 pages!) and he is editing it, to be released in batches. The first part is now available, on the arXiv as This Week’s Finds in Mathematical Physics (1-50). As I find time, I’m looking forward to reading through these, encourage everyone interested in math and physics to do the same.
  • Frank Wilczek has a new book out, and there’s an interview with him at Quanta. You can see a conversation between him and Brian Greene here on Friday.
  • Another physicist with a new book is Jesper Grimstrup, whose Shell Beach: The search for the final theory I’ve just finished reading and enjoyed greatly. The book is quite personal and non-technical, with topic Grimstrup’s life as a theorist pursuing a unified theory. His career story is quite interesting, giving insight into the ways academic theoretical physics is challenging for young theorists trying to pursue non-mainstream research programs. Several books have appeared in recent years aimed at putting this kind of physics research in a human and philosophical context, telling you what it has to do with the meaning of life. There’s some of that in this book too, of a much more compelling sort than what you see elsewhere. Grimstrup has a website here, and in recent years has ended up leaving academia and trying to fund his research with donations. I can think of a lot worse things you could do with your money than send him some.

    I’m quite sympathetic to the underlying theme that he describes pursuing (together with Johannes Aastrup) in the book, that of bringing together the insights of loop quantum gravity and non-commutative geometry. More recently they’ve been working on some new ideas for formulating QFT non-perturbatively that seem worth investigating. There’s a survey blog post here.

Update: Another bit of private math/physics funding news. The IAS has announced establishment of the Carl P. Feinberg Cross-Disciplinary Program in Innovation

Scientific research at the Institute is traditionally driven by the collaboration and independent projects of a full-time Faculty and a revolving class of more than 200 researchers at various stages in their careers. The Carl P. Feinberg Cross-Disciplinary Program in Innovation will build on this successful model with the recruitment of mid-career scholars who have pioneered foundational developments in new areas. Bringing together scholars with such unique insights—which may not be obviously connected to the existing themes of the past 20 or 30 or 40 years—ensures that IAS will remain agile and responsive to new intellectual developments that do not yet fit the mold of what graduate students and postdocs generally know. In order to close this knowledge gap, the program will feature intense, focused workshops and “master classes.”

“Since its founding, the Institute has served as a world center for investigations into the fundamental laws of nature. We are currently in the middle of a grand symbiosis of ideas, from the equations of general relativity to the quantum information of black holes,” stated Robbert Dijkgraaf, IAS Director and Leon Levy Professor. “This revolutionary program will provide a dedicated space and the necessary flexibility to accelerate these exciting developments, and will surely forge new connections across fields.”

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Martin Veltman 1931-2021

I heard today of the recent death of Martin Veltman, a theorist largely responsible (with his student Gerard ‘t Hooft) for showing the renormalizability of non-abelian gauge theories, a breakthrough crucial to the Standard Model that won both of the them the 1999 Nobel Prize. For the story of this work, the best source is likely Veltman’s Nobel lecture.

My one memory of meeting Veltman in person was when he visited Stony Brook at the time that I was a postdoc there (mid 1980s). There was a party at someone’s house, and I spent part of the evening talking to him then. What most struck me was his great passion for whatever it was we were talking about. One topic I remember was the computer algebra program Schoonschip (which Wolfram acknowledges as an inspiration for Mathematica). I vaguely recall that at that time Veltman had recently ported the program to a microprocessor and he was selling copies in some form. It also seems to me that one remarkable aspect of the program was that it was written in assembly language, not compiled from a higher level language. At the time I was doing computer calculations, but of a very different kind (lattice gauge theory Monte-Carlos). Since my own interests were focused on non-perturbative calculations, I wasn’t paying much attention to Veltman’s work, although I do remember finding his Diagrammar document (written with ‘t Hooft) quite fascinating.

A comment that evening that really struck me was about students, in particular that “you give your students your life-blood!”. This seemed likely to have some reference to Veltman’s relations with his ex-student ‘t Hooft, but I’m pretty sure I didn’t quiz him on that topic.

Many years later, when I was trying to get Not Even Wrong published, I contacted Veltman and he was quite helpful. At the time he had recently published his own popular book about particle physics, Facts and Mysteries in Elementary Particles, which contained his own version of the Not Even Wrong critique:

The reader may ask why in this book string theory and supersymmetry have not been discussed. . . The fact is that this book is about physics and this implies that theoretical ideas must be supported by experimental facts. Neither supersymmetry nor string theory satisfy this criterion. They are figments of the theoretical mind. To quote Pauli, they are not even wrong. They have no place here.

That book is quite good, I strongly recommend it. May its author rest in peace.

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DAU. String Theory

I first wrote here in 2015 about DAU, the unusual film project based to some extent on the life of Landau. Parts of the film first were shown in Paris early in 2019, and this past year started appearing on the DAU website. I’d been looking forward to seeing Gross, Yau, Rovelli and others in the film, so paid to watch one of the first parts, DAU. Degeneration, when it became available last year. It’s over six hours long, for a review, see here. I ended up doing a certain amount of fast-forwarding, was disappointed to only see Nikita Nekrasov and Dmitri Kaledin, none of the other math/physics world figures I had heard had participated.

DAU largely was funded by Russian oligarch Sergei Adoniev. For an excellent article discussing the project and its context in current Russian culture, see Sophie Pinkham’s article Nihilism for Oligarchs.

There wasn’t much physics in DAU. Degneration, but evidently it plays a significant role in other parts of the film. According to the DAU website,

Real-life scientists, who were able to continue with their research in the Institute, included: physicist Andrei Losev; mathematicians Dmitri Kaledin and Shing-Tung Yau; string theorist Nikita Nekrasov; Nobel-Prize winning physicist David Gross; neuroscientist James Fallon; and biochemist Luc Bigé. “One group was researching string theory and another researching quantum gravity. These groups hated each other. One stated there were 12 dimensions, the other claimed there were 24. The string theory group believed there couldn’t be 24 dimensions. The quantum gravity group believed that the other scientists were narrow-minded,” explained Khrzhanovskiy.

Now available is a part which seems to more centrally involve physics, DAU. String Theory, which is described as follows:

Nikita Nekrasov is a scientist, a theoretical physicist who studies our world and other possible worlds. He refuses to make a choice between mathematics and physics, between one woman and another, as he ponders the existence of the multi-universe. At scientific conferences, attended by eminent foreign scientists and a rising younger generation of physicists alike, Nekrasov gets carried away debating the beauty of string theory. He attempts to explain to all of his women – Katya, the librarian, Zoya, the scientific secretary, Svetalana, the head of department – about the theory of his own polygamy, and the possibility of having enough feelings to satisfy everyone.

Multiple universes have always been advertised with “in some other universe you’re dating Scarlett Johansson”, relating the idea to multiple partners in this universe is an innovation.

I haven’t yet watched DAU. String Theory, will likely find time for that soon. I’m worried that I’ll still not get to see Gross, Yau, Rovelli and others though, and lack the time and energy to look through all the other parts of the film. I’d like to crowd-source a solution to this problem: if anyone watching these things can let the rest of us know in which parts (at what times) well-known math/physics personalities appear, that would be greatly appreciated.

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