A New Approach to Modularity

I was assuming that Peter Scholze’s Emmy Noether lectures at the IAS would be the big news about advances in the Langlands program this coming week, but an anonymous correspondent just sent me this link. Tomorrow Andrew Wiles will be giving a talk in Oxford on “A New Approach to Modularity”, with abstract:

In the 1960’s Langlands proposed a generalisation of Class Field Theory. I will review this and describe a new approach using the trace formua as well as some analytic arguments reminiscent of those used in the classical case. In more concrete terms the problem is to prove general modularity theorems, and I will explain the progress I have made on this problem.

I’m curious to hear from anyone who knows what this is about or can report tomorrow after the talk. Wiles does have a certain track record of unveiling unexpected huge progress in a talk like this…

Update: Well, the talk should be over now. Sure someone who was there can let us know what happened?

Update: There’s a tweet with some photos.

Posted in Langlands | 3 Comments

Three Items

Kind of like the last posting, but this time you get two worthwhile items to make up for one that’s not.

  • Dan Garisto has a very good article here examining the present state of high energy experimental particle physics and phenomenology. He also summarizes current thoughts about the future. The CERN FCC-ee proposal is still in feasibility study mode, with the big problem its high cost. Numbers like \$15 – \$20 billion have shown up in press reports, and Garisto has “tens of billions”. The feasibility study is supposed to be finished late next year, and presumably a big part of it is people at CERN crunching numbers trying to figure out some plausible way this could work financially.
  • There’s a very good article at Quanta about efforts to put together version 3 of the very influential “Kirby List” of open problems in topology. For version 1, see here, version 2 here, and as far as I can tell, version 3 still not finished (but discussed here).
  • Brian Keating has a video asking “What would It take for String Theory to move beyond the realm of pure math, Into verifiable territory”? The answer obviously is a conventional substantive scientific prediction of anything. By describing the problem with string theory as being that it is now in “the realm of pure math”, I think Keating is missing the point. The problem with “string theory” is not that it’s pure math, but that it’s not a theory. “String theory” is now a 50 year old set of failed hopes and dreams that a theory might exist, see for instance Theorists Without a Theory. Every so often I try to figure out what people still pursuing this are up to, most recently today taking a quick look at KITP talks by Liam McAllister here and here.

    The problem shows up clearly at 10:23 of the first talk where he’s talking about his goal, which gives up on studying all or even representative string theory solutions and tries just to find any “valid solutions”. But what is a “valid solution”?

    And by valid solutions, we should say what equations are we trying to solve? And, it’s not the case that we can currently think about finding cosmological solutions of the exact theory in any non-perturbative sense.

    McAllister artfully avoids saying what everyone at the talk knows (after all, the talk is part of a program entitled “What is String Theory?”): you can’t look for solutions to the theory because there is no theory. More specifically, no “exact” theory, just a long list of possible theories that one hopes might be in some sense approximations to a real theory. He then goes on to specify the extremely complex approximate theory he wants to work in, chosen by the “look under the lamppost” method as something you could actually imagine calculating. Whenever I look at things of this kind I’m completely mystified why anyone thinks it make sense to embark on insanely complicated calculations like these with essentially zero credible scientific motivation. Somehow though, he has a whole group of people doing this. By the end of the second talk he’s giving his vision of the future, which features an old photo of a room of hundreds of men in suits and ties calculating with pencil, paper and slide rules.

    I was mystified twenty years ago why anyone thought this was a good idea, and the whole thing has just gotten stranger and stranger…

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Two Items

If I’m going to point to something about string theory and say the same things as always about it, seems best to first start with the opposite, an item about something really worth reading.

  • This spring I’ve been teaching a graduate course aimed at getting to an explanation of the Standard Model aimed at mathematicians. The first few weeks have been about quantization and quantum mechanics, today I’m starting on quantum field theory, starting with developing the framework of non-relativistic QFT. While trying to figure out how best to pass from QM to QFT, I’ve kept coming across various aspects of this that I’ve always found confusing, never seen a good explanation of. Today I ran across a wonderful article by Thanu Padmanabhan, who I knew about just because of his very good introductory book on QFT, Quantum Field Theory: The why, what and how. The article is called “Obtaining the Non-relativistic Quantum Mechanics from Quantum Field Theory: Issues, Folklores and Facts” and subtitled “What happens to the anti-particles when you take the non-relativistic limit of QFT?” It contains a lot of very clear discussion of issues that come up when you try and think about the QM/QFT relationship, a sort of thing I haven’t seen anywhere else.

    Looking for more of Padmanabhan’s writings, I was sad to find out that he passed away in 2021 at a relatively young age, which is a great loss. For more about him, there’s a collection of essays by those who knew him available here.

  • For something I can’t recommend paying more attention to, New Scientist has an article labeled How to Test String Theory, which is mostly an interview with Joseph Conlon. Conlon’s goal is to make the case for string theory, in its original form as a unified theory with compactified extra dimensions. On the issue of testability there’s nothing new, just the usual unfalsifiable story that among all of the extra stuff (moduli fields, axions, extra dimension, extended structures) that appears in string theory and that string theorists have to go to great trouble to make non-observable, it’s in principle conceivable that somebody might observe one of these things someday. But, that’s not really what people mean when they ask for a test. For details of the sort of thing he’s talking about in the New Scientist article, see here.

    I strongly disagree with Conlon about some of what he’s saying, but the situation is very much like it has always been with many string theorists since way back to nearly forty years ago. We don’t disagree about the facts, it’s just that I’ve always looked at these facts and interpreted them as showing string theory unification ideas to be unpromising, whereas string theorists like Conlon somehow find reasons for optimism, or at least for believing there’s no better thing to do with their time. Last time I was in Oxford, Conlon invited me to lunch at his College and I enjoyed our conversation. I think we agreed on many topics, even about what is going on in string theory, but it looks like we’re always going to have diametrically opposed views on this particular question. For more from him, as far as popular books by string theorists go, his Why String Theory? book is about the best there is, see more about this here.

Update: A couple more.

  • Curt Jaimungal has a conversation with Lee Smolin.
  • Andy Strominger is giving talks on Celestial Holography at the KITP “What is String Theory?” program, first one is here. Lots of questions from the audience. One thing he makes clear is that this is not leading to a theory of quantum gravity. Stringking is back, his comments on this:

    KITP program on what is string theory such a joke this week. Strominger shilling his celestial vaporware.

Posted in This Week's Hype | 13 Comments

Why physicists are rethinking the route to a theory of everything

New Scientist this week has a cover story I can strongly endorse, entitled Why physicists are rethinking the route to a theory of everything. It’s by journalist Michael Brooks, partly based on a long conversation we had a month or so ago. Unfortunately it’s behind a paywall, but I’ll provide a summary and some extracts here.

For a more technical description of the ideas I’m pursuing that are discussed in the article, see this preprint, which was intended to be a very short and concise explanation of what I think is the most important new idea here. This semester I’m mainly working on teaching and writing up notes for an advanced course for math graduate students about the Standard Model. I’ll be adding to these as the semester goes on, but have just added preliminary versions of two chapters (9 and 10) about the geometry of vectors, spinors and twistors in four dimensions. These chapters give a careful explanation of the standard story, according to which spacetime vectors are a tensor product of left-handed and right-handed spinors. They don’t include a discussion of the alternative I’m pursuing: spacetime vectors are a tensor product of right-handed spinors and complex conjugated right-handed spinors. I’ll write more about that later, after the course is over end of April (and I’ve recovered with a vacation early May…).

It’s well-known to theorists that the Standard Model theory is largely determined by its choice of symmetries (spacetime and internal). A goal of these notes is to emphasize that aspect of the theory, rather than the usual point of view that this is all about writing down fields and the terms of a Lagrangian. This symmetry-based point of view should make it easier to see what happens when you make the sort of change in how the symmetries work that I’m proposing. What I’m not doing is looking for a new Lagrangian. All evidence is that we have the right Lagrangian (the Standard Model Lagrangian), but there is more to understand about its structure and how its symmetries work. In particular, the different choice of relation between vectors and spinors that I am proposing is not different if you just look at Minkowski spacetime, but is quite different if you look at Euclidean spacetime.

The New Scientist article has an overall theme of new ideas about unification grounded in geometry:

… a spate of new would-be final theories aren’t grounded in physics at all, but in a wild landscape of abstract geometry…

That might strike you as outlandish, but it makes sense to Peter Woit, a mathematician at Columbia University in New York. “Our best theories are already very deeply geometrical,” he says.

There’s some discussion of string theory and its problems, with David Berman’s characterization “It can be a theory of everything, but probably it’s a theory of too much.” The article goes on to describe the amplituhedron program, with quotes from Jaroslav Trnka. After noting that it only describes some specific theories, there’s

Trnka thinks the amplituhedron approach might enable us to go even further. “One can speculate that whatever the correct theory of everything is, it would be naturally described in the amplituhedron language,” he says.

Turning to a discussion of twistors, there’s then a section describing my ideas fairly well:

Woit is using spinors and twistors to create what he hopes are the foundations of a theory of everything. He describes space and time using vectors, which are mathematical instructions for how to move between two points in space and time – that are the product of two spinors. “The conventional thing to do has been to say that space-time vectors are products of a right-handed and a left-handed spinor,” says Woit. But he claims he has now worked out how to create space-time from two copies of the right-handed spinors.

The beauty of it, says Woit, is that this “right-handed space-time” leaves the left-handed spinors free to create particle physics. In quantum field theory, spinors are used to describe fermions, the particles of ordinary matter. So Woit’s insights into spinor geometry might lead to laws describing the holy trinity of space, time and matter.

The idea has got Woit excited. He has spent most of his career looking at other ideas, thinking they will go somewhere, and being disappointed. “But the more I looked at twistor theory, the more it didn’t fall apart,” he says. “Not only that, I keep discovering new ways in which it actually works.”

It isn’t that Woit believes he necessarily has the answers. But, he says, it is good to know that, despite the long search for a theory of everything, there are still new possibilities opening up. And a better, if not perfect, theory has to be out there, he reckons, one that at least deals with sticking points like dark matter. “If you look at what we have, and its problems, you know you can do better,” he says.

Other work that is described is Renate Loll and causal dynamical triangulations, as well as what Jesper Grimstrup and Johannes Aastrup call “quantum holonomy diffeomorphisms.” For more details of the Aastrup/Grimstrup ideas, see this preprint as well as Grimstrup’s website. I can also recommend his memoir, Shell Beach.

The article ends with:

That said, when Woit – who has long been known as an arch cynic – is excited about the search for a theory of everything again, maybe all bets are off. Playing with twistors has changed him, he says. “I’ve spent most of my life saying that I don’t have a convincing idea and I don’t know anyone who does. But now I’m sending people emails saying: ‘Oh, I have this great idea’.”

Woit says it with a grin, acknowledging the hubris of thinking that maybe, after so many millennia, we might finally have cracked the universe open. “Of course, it may be that there’s something wrong with me,” he says. “Maybe I’ve just gotten old and just lost my way.”

Posted in Euclidean Twistor Unification | 50 Comments

Arithmetic, Geometry and QFT news

This week at Harvard’s CMSA there’s a program on Arithmetic Quantum Field Theory that is starting up and will continue through March. There’s a series of introductory talks going on this week, by Minhyong Kim, Brian Williams, and David Ben-Zvi. I believe video and/or notes of the talks will be made available.

At the IHES and the Max Planck Institute, the Clausen-Scholze joint course on analytic stacks has just ended. For an article (in German) about them and the topic of the course, see here. What they’re working on provides some new very foundational ideas about spaces and geometry, in both the arithmetic and conventional real or complex geometry contexts. Many of the course lectures are pretty technical, but I recommend watching the last lecture, where Scholze explains what they hope can be done with these new foundations.

Of the applications, the one that interests me most is the one that was a motivation for Scholze to develop these ideas, the question of how to extend his work with Fargues on local Langlands as geometric Langlands to the case of real Lie groups. He’ll be giving a series of talks about this at the IAS next month.

Something to look forward to in the future is seeing the new Clausen-Scholze ideas about geometry and arithmetic showing up in the sort of relations between QFT, arithmetic and geometry being discussed at the CMSA.

Posted in Langlands | 10 Comments

This Week’s Hype

For the last thirty years or so, one tactic of those who refuse to admit the failure of string theory has been to go to the press with bogus claims of “we finally have found a way to get testable predictions from string theory!”. I’ve written about dozens and dozens of these over the years (see here). In recent years the number of these has tapered off considerably, as it likely has become harder and harder to find anyone who will take this seriously, given the track record of such claims.

Today though, Quanta magazine has a new example, with an article that informs us

An idea derived from string theory suggests that dark matter is hiding in a (relatively) large extra dimension. The theory makes testable predictions that physicists are investigating now.

This is about a proposal for a micron-scale large extra dimension, with no significant connection to string theory. I took a look at the “predictions” (see here) long enough to assure myself it’s more of the same, better to not spend more of one’s time on it. One positive thing to say about the article is that the writer did go ask string theorist experts about this, and while these experts tried to be polite, they clearly weren’t enthusiastic:

While physicists find the dark dimension proposal intriguing, some are skeptical that it will work out. “Searching for extra dimensions through more precise experiments is a very interesting thing to do,” said Juan Maldacena, a physicist at the Institute for Advanced Study, “though I think that the probability of finding them is low.”

Joseph Conlon, a physicist at Oxford, shares that skepticism: “There are many ideas that would be important if true, but are probably not. This is one of them. The conjectures it is based on are somewhat ambitious, and I think the current evidence for them is rather weak.”

Better though would have been to ask Sabine Hossenfelder what she thinks about this kind of thing (or not write about them at all)…

Update: Vafa has a new paper explaining the “prediction” of the extra dimension from Swampland conjectures. According to him

The most direct way to test the dark dimension scenario is to check Newton’s gravitational inverse square law (ISL) at micron scale. Due to O(1) number ambiguities one can only predict this to appear at length scales 1 − 10 microns. Experiments checking this length scale would need to improve the current range bounds by a factor of 10.

What if such experiments can be done but don’t see anything, even down to 1/10,000th of a micron? No problem at all, that would be a new discovery that you need to change one of the many Swampland conjectures:

We can only wait for the experimental verdict. Either way, we will learn exciting new physics!

Update: Sabine Hossenfelder explainer here.

Posted in This Week's Hype | 31 Comments

Particle Physics Is Not In Crisis

For a low-rent version of the self-congratulatory program discussed here, Bad Boy of Science Sam Gregson has a new video up entitled Particle Physics Is Not In Crisis – but we can make improvements. Cliff Burgess plays the Strominger role, explaining that the idea that there’s any problem with what’s going on in particle theory is “a nothing-burger” and “a complete non-issue”. Asked to rank any such problem on a scale of 0-10, he gives the Strominger-esque “.0001”. Martin Bauer goes for “1”.

The take on the question is much the same as Sean Carroll’s four-hour plus explanation that there is no problem, but shorter. It’s similar to Carroll in that no one who thinks there is a problem was invited to participate, or even gets mentioned by name. There’s a repeated reference to mysterious “Twitter influencers”, which I find very confusing because just about the only particle theorists I see spending time on Twitter going on about the state of the field are Bauer and Burgess. They can’t mean me since I’ve so far resisted the temptation to enter Twitter discussions. The idea of trying to have a serious discussion of complex scientific issues in the Twitter format never made any sense to me, and (StringKing aside) I find it hard to think of any tweets by anyone that shed any light on serious issues in this area.

The more serious part of the program was the discussion among the two HEP experimentalists of the state of their field, which got a 5-6 on the crisis level scale. I wrote about the problem there five years ago, and very little has changed, other than that we’re five years closer to the date when there will no longer be an energy frontier machine running anywhere in the world. The underlying problem wasn’t really explained. CERN is working on it, but there is as of now no specific plan with specific budget numbers for what to build next. Maybe I misunderstood, but it seemed that Bauer and others were talking about how the field just needed to convince funding agencies to support budget numbers of order \$100 billion, which is a pipe dream.

Update: Latest podcast from Sean Carroll has nothing to do with the crisis in particle physics, but he starts off anyway with this:

You may have heard there is a crisis in physics. No, there’s not. I mean, there’s little tiny crises, but that’s the very standard procedure if you’re doing science at the cutting edge, is all sorts of puzzles that we don’t know the answer to.

“Little tiny crises” is I guess his version of the Cliff Burgess “.0001” and Andy Strominger “A+++”.

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Quick Links

A few quick links:

  • There’s a one-day conference next Friday at the IHES, recognizing Dustin Clausen’s appointment to a new Jean-Pierre Bourguignon Chair. Should be several interesting talks, see here.
  • There’s an ongoing conference at the KITP on the topic of What is String Theory? So far, none of the online talks address that issue. Evidently there was a discussion of the topic last Wednesday, but not recorded. Were any readers here in attendance and willing to report on that event? Next chance to find out what string theory is will be a Monday Blackboard Lunch talk by Gopakumar.
  • In April there will be an IUT conference hosted by Zen University in Tokyo, see here. All the speakers but one are from RIMS. For news from the senior people devoted to IUT, Ivan Fesenko has moved to Westlake University in Hangzhou, and Shinichi Mochizuki is has been blogging here.
  • There’s a new Shanghai Institute for Mathematics and Interdisciplinary Science, headed by Shing-Tung Yau.
  • For those following what happens with the small number of permanent positions in particle theory, news from 4 gravitons.

Update: One more, which I’m quite interested in. Scholze will be giving a series of three Emmy Noether lectures at the IAS in March, topic Real local Langlands as geometric Langlands on the twistor-P1.

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Know Time Podcast

For another podcast/interview with me that was recently recorded, see Maths, Twistors & String Theory. Know Time is a series of podcasts that is a project of Shalaj Lawania, and I was impressed by the effort he put into trying to make sense of a complicated and inaccessible subject. For an excellent pairing with what I have to say, see his earlier interview with Matthew Kleban, who has a more positive take on string theory, the multiverse, etc.

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Spring Course

Starting next week I’ll be teaching a graduate topics course, with the general plan to develop much of the quantum field theory of the Standard Model in a form accessible to mathematicians, emphasizing the connections to representation theory. There’s a course web-page here, notes will start appearing here once the course gets underway. While the course will be aimed at mathematicians, I’m hoping that some physicists might find it interesting and worth trying to follow.

The last time I did something like this was back in fall 2003. At that time the course was aimed at getting math students to the point of understanding the TQFTs for Chern-Simons theory and Donaldson theory and was very much based on the path integral. This time I’ll be mostly sticking to flat space-time and using more representation theory. Also, a lot more about spinor geometry, as well about about how Euclidean and Minkowski space-time versions of QFT are related.

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