Not Even Wrong

Yet more math items:

• First of all, congratulations to my colleague Johan de Jong, recipient of the 2022 AMS Steele Prize for Mathematical Exposition. Johan’s Stacks Project is very much deserving of such recognition. It’s both huge in scale and very high in quality, with nothing else really comparable. While it has attracted many contributors, it has always been mostly a one-person effort. If you’re interested in helping, even those not so expert in the field can contribute by fixing any mistakes they might find when using this incredible resource.
• On my currently favorite topic of the unity of math (and physics), there’s a talk by Barry Mazur, in which he begins by raising the question “What is it that unifies Mathematics?”. He goes on to turn around the question “What is the physical interpretation of the Jones polynomial?” asked by Atiyah (and answered by Witten’s Chern-Simons theory). Mazur asks:
  

  What is the Arithmetical-Algebraic-Geometric interpretation of the Jones polynomial?

  or of Chern-Simons theory?

  or of TQFT?

• Mazur’s title is “Bridges between Geometry and Number Theory”. The metaphor of “bridges” to describe what unifies mathematics gets a workout in a recent Quanta article about Ana Cariani and the Langlands program entitled The Mathematician Who Delights in Building Bridges (and subtitled Ana Caraiani seeks to unify mathematics through her work on the ambitious Langlands program.)
• At the same conference as the one with the Mazur talk, Maxim Kontsevich spoke on Geometry from the perspective of quantum mechanics and string theory. His talk was a great summary of various aspects of the problem of quantization, in both quantum mechanics and conformal field theory. There wasn’t much though about what has been going on since the early developments in conformal field theory that he discussed. Things got a bit worrisome at the end, when he announced that he can’t understand Kevin Costello these days (if he can’t, who can?), and ended with (here’s a google-aided transcript):
  

  You see that gauge theories and gravity appears in various interactions is it’s in nothing else in a sense, and geometric limits of various string theories or quantum field theories and what I claim that it’s in fact it’s something generally about complex systems and mathematics. You do some combinatorial problem, whatever it is you get some counting or something, and then maybe you look on asymptotic growth of the number of solutions. It could be something very simple but your arranged parameters became something more complicated and if you see something more complicated it’s kind of I think it’s unavoidable you see some physics in a very wide sense: some string theory, some membranes, whatever. Okay, thank you.

  I can’t really make much sense of this, but he seems to have some sort of vision of fundamental physics being linked with complexity, a point of view that seems increasingly common, while not leading anywhere promising.

Moving to purely physics topics:

• Noah Miller was a student here at Columbia in one of my mathematics of QM courses. I’ve had some wonderful students in those classes, and he was one of the best. He has gone on to graduate study in physics at Harvard, and I just saw a beautiful new paper by him this week on the arXiv, From Noether’s Theorem to Bremsstrahlung: a pedagogical introduction to large gauge transformations and classical soft theorems. It’s an exposition for non-experts of some of the new ideas about gauge symmetry and physics that Strominger and collaborators have been working on, highly lucid and readable.
• I very much recommend taking a look at the talk from earlier this year by Mikhail Shaposhnikov, Conformal symmetry: towards the link between the Fermi and the Planck scales. Shaposhnikov has done a lot of fascinating work over the years, developing in detail a point of view which hasn’t got a lot of attention, but that seems to me very compelling. He argues that the SM and GR make a perfectly consistent theory up to the Planck scale, with the “naturalness problem” disappearing when you don’t assume something like a GUT scale with new heavy particles. Watching the discussion after the talk, one sees how many people find it hard to envision such a possibility, even though all experimental evidence shows no signs of such particles. For more about what he is in mind, see the talk or some of the many papers he’s been writing about this.
• Finally, skydivephil tells me he has managed to get David Gross and Carlo Rovelli to debate string theory vs. loop quantum gravity, with video to drop on Youtube tomorrow. I normally try to make it a policy to avoid getting into this particular debate, but this I have to see. While you’re waiting for this, you can watch an earlier pairing well worth seeing: Alan Guth and Roger Penrose debating the multiverse versus cyclic cosmology.

Update: I just watched the Gross/Rovelli debate, and thought Rovelli did a good job of making the case that string theory is a failed research program. Gross spoke uninterruptedly at length, but interrupted Rovelli constantly. I found it interesting that Gross acknowledged “supersymmetry hype” and hype back in 1984-5, while at the same time engaging in massive amounts of hype about the current state of string theory. On the time scale for progress in string theory, he says 80 years (end of the century) to understand how to use string theory to solve QCD, no time scale for getting unification out of string theory.

Gross’s main point he kept repeating is that “string theory” now means an overarching framework that includes the Standard Model, so there’s no distinction between the Standard Model and “string theory” and you can’t argue that “string theory” is a failure. This argument is so silly that it’s hard to engage with it in any sensible way, and Rovelli didn’t even try.

Update: There’s an interesting long interview with Andy Strominger here. Some of this brought back old memories, since Strominger overlapped with me a bit as an undergraduate at Harvard, although the story of that part of his life is very unusual. I hadn’t realized the extent to which from the very beginning he was focused on the problem of quantum gravity, which to some extent explains his lack of interest in particle physics extensions of the Standard Model.

One thing he makes clear is that at this point string theory has become completely disconnected from the possibility of saying something testable about the real world. The AIP interviewer kept trying to ask about that, leading to this exchange:

Zierler: So is there an experiment that you can conceive of that could disprove string theory?

Strominger: I guess I am not getting my point across.

Zierler: You’re saying that string theory is totally outside the world of experimentation.

Strominger: … So yes, I don’t think – not many string theorists will talk this way – but I don’t think that we are in my lifetime — and I’m planning to live a very long time — going to get direct experimental evidence for string theory.

On the issue of what the terms “string theory” now mean. Strominger makes it clear that from the point of view of him and many others, there’s no longer any possible critique of “string theory” as a fundamental physical theory:

For the last 30 years, everything new that we’ve discovered, as long as we can relate it to the ideas in string theory, we call it string theory. So, if we continue to call everything that we discover string theory, it’s virtually certain that– (both laugh) It’s certain that when we get to the answer, we’ll call it string theory!

First some math news:

• An anonymous commenter claims here that the 2026 ICM will take place in Philadelphia. I had heard that a US group was submitting a proposal, so this rumor is plausible.
• Many mathematicians and physicists have signed an Open Letter on K-12 Mathematics pointing to problems with attempts to reform mathematics education such as the California Mathematics Framework. For more about this, see the blog entry posted here and on Scott Aaronson’s blog, and more detail here.

  While I’ve always had some sympathy for the general idea that there’s much that could be changed and improved about the US K-12 math curriculum, there’s a huge problem with all proposed changes based on the “algebra/pre-calculus/calculus sequence is too hard and not relevant to everyday life” argument. Students leaving high school without algebra and some pre-calculus are put in a position such that they’re unequipped to study calculus, and calculus is fundamental to learning physics. Without being able to learn physics, a huge range of possible fields of study and careers will be closed to them, from much of engineering through even going to medical school. Whatever change one makes to K-12 math education, it shouldn’t leave students entering college with a severely limited choice of fields they are prepared to study.

• Davide Castelvecchi at Nature has a story about machine learning being useful in knot theory and representation theory. Given my personal prejudice that hearing endlessly about how AI and machine learning will take over everything is just depressing, I’m trying to ignore this kind of thing. But, together with stories like the success of proof assistants in solving a problem posed by Scholze, it’s harder and harder to believe what I would like to believe (that this is all a bunch of hype that should be ignored).

For some physics items:

• Jim Baggott has an excellent article at Aeon about the “Shut up and calculate” meme, featuring a retraction by its originator, David Mermin

  In a quick follow-up discussion with me in July 2021, Mermin confessed that he now regrets his choice of words. Already by 2004 he had ‘come to hold a milder and more nuanced opinion of the Copenhagen view’. He had accepted that ‘Shut up and calculate’ was ‘not very clever. It’s snide and mindlessly dismissive.’ But he also felt that he had nothing to be ashamed of ‘other than having characterized the Copenhagen interpretation in such foolish terms’.

• For some wisdom on the thorny issue of how to relate Euclidean and Minkowski signature metrics in gravity, see the recent IAS lecture by Graeme Segal on Wick Rotation and the Positivity of Energy in Quantum Field Theory.
• In fundamental theoretical physics these days, it’s quantum information theory all the time, with conferences around now here, here, here, and here. I can’t figure out what the relevance of any of this is supposed to be to actual models describing reality. Best guess would be that this is supposed to “solve the black hole information loss paradox”, although in that case Sabine Hossenfelder has some apt comments here.
• For something more inspirational, see Natalie Wolchover’s long piece at Quanta on the JWST.

Update: For more from Geordie Williamson about the math/AI story, see here. For more about the problems with the California Mathematics Framework and its co-author see here.

A couple months ago I recorded a podcast with Lex Fridman, it’s now available here.

A lot of Fridman’s other interviews are well worth watching or listening to, and I thought we had an interesting conversation. I can’t stand listening to or watching myself, so not sure how it turned out. But happy to answer here any questions about what we were discussing.

During recent travels I attended two conferences (in Paris and Berkeley) and met up with quite a few people. At the Paris conference I gave an intentionally provocative talk to the philosophers of physics there, slides are here. The argument I was trying to make is essentially that more attention should be paid to evidence for a deep unity in much of modern mathematics, which at the same time is connected to our best unified theory of physics (the Standard Model and GR). Edward Frenkel has made some similar points, referring to the Langlands program and its connections to physics as a “Grand Unified Theory of Mathematics”. The specific structures underlying this unification seem to me to deserve attention as providing an important way of thinking about what’s at the “foundations” of both math and physics.

Another motivation for this talk was to make an argument against what I see as having become a widespread and standard ideology about the search for a unified theory in physics. Talking to many physicists and mathematicians interested in physics, I noticed that the conventional wisdom, shared by the establishment and contrarians alike, is that the SM and GR are likely low energy emergent theories, that some completely different sort of theory is needed to describe very short distances such as the Planck scale. Physics establishment figures tend to believe that following the path started with string theory, then AdS/CFT, lately quantum error correction or whatever, will someday lead to a dramatically different sort of theory, replacing space, time and maybe quantum mechanics. Contrarians often have their own favorite idea for a radically different starting point. For an example of this, take a look at Figures 2 and 3 of Mike Freedman’s The Universe from a Single Particle (he spoke about this in Berkeley). Figure 2 is the “establishment” picture, with AdS/CFT the fundamental theory, well-decoupled from the emergent SM + GR (since no one has any idea how to relate them). His Figure 3 shows his own proposal, even better decoupled from any connection to the SM + GR.

Given the extreme level of experimental success of the SM + GR, the obvious conjecture is that these are close to a unified theory valid at all distances. That the mathematical framework they are built on is closely connected to unifying structures in mathematics provides yet more evidence that what one is looking for is not something completely different. The odd thing about the present moment is that arguing that our well-established successful theories can provide a solid basis for further unification makes one a contrarian, with the “establishment” position that a revolution sweeping such theories aside is needed.

I hope to find time in the next few weeks to write up what’s outlined in the slides as a more detailed article of some sort. More immediately, I plan to write a blog entry and perhaps some more detailed notes about the “twistor $P^1$” mentioned at the end of the talk, explaining how it shows up in Euclidean twistor theory as well as in recent work on the Langlands program.

First some personally relevant items:

• I finally have a finished version 2.0 of my euclidean twistor unification paper (discussed here), it’s uploaded to the arXiv, should appear there Monday.
• I’ll be giving a talk October 30th at the Foundations 2021 conference in Paris, something about the unity of math and physics, and will take the opportunity to spend about two weeks in Europe, mostly in Paris.
• In other travels, I’ll be in the Bay area for a week or so mid-November.
• For entertainment I tried out a WordPress plugin that dumps all my blog content into a single pdf. If you want 8,518 pages to read at your leisure when you’re not connected to the internet, this would be one way to spend your time.

In math and physics news, there’s:

• David Mumford has a had a remarkable career, first as an algebraic geometer (he won a Fields Medal for his work in this area) and later in the field of computer vision. He’s also known as a talented expositor, with his books and papers the standard references for several different topics. He’s moved into physics this month, with a wonderful article about cosmology in the Notices. His blog is well-worth following, it had the cosmology piece a few months back.
• Also in the Notices is a set of memorial articles about Lucien Szpiro, who passed away last year. I wrote a little bit about him here, am very pleased to see these articles which give a detailed picture of both the person and his mathematics.
• The Simons Collaboration on Global Categorical Symmetries had its kick-off meeting this week in Stony Brook, videos available here. There are many interesting talks to watch. I got very excited for a minute (around :05:00 in this video) when Greg Moore started talking about some of my favorite questions (e.g. what is the representation theory of gauge groups in dimension greater than one?). But then I realized he had labeled these “Traditional Questions”, in Fraktur font to emphasize how old and out of date they were. He described these as “old-fashioned questions”, that people were not seriously working on anymore. As he explained, you’re no longer supposed to be thinking about a fixed topology, but looking for something more general that treats all topologies. My problem with this is that one tends to get interesting results about topology this way, but the physics applications seem to be in condensed matter physics, with little relevance to questions about local fundamental physics that have always been my main interest.
• I really don’t understand the thinking in physics theses days at all. Nima Arkani-Hamed is a remarkable theorist who came up with a lot of highly speculative ideas about particle physics that have never worked out, then moved on to brilliant work leading efforts that have transformed the study of scattering amplitudes. The APS just announced that he’s getting the 2022 J. J. Sakurai Prize for Theoretical Particle Physics, for “the development of transformative new frameworks”. These “transformative new frameworks” are listed as “work on large extra dimensions, the Little Higgs, and more generally for new ideas connected to the origin of the electroweak scale”, none of which has had any success, while the amplitudes work is ignored.

I just noticed that Gordon Kane has recently published a second edition of his 2017 String Theory and the Real World. Columbia doesn’t seem to yet have full online access to the second edition, but one can already compare the two editions in a few places. For instance, on page 1-5 of the 2017 edition one reads

The LHC is now working in a region of energy and intensity where well-motivated theories imply superpartners could be seen by late 2018.

and

There is good reason, based on theory, to think discovery of the superpartners of Standard Model particles should occur at the CERN LHC in the next few years.

The corresponding first chapter of the latest edition has:

The LHC has so far just entered the region of superpartner masses predicted by compactified theories, which ranges from about 1.5 to ∼5 GeV (we’ll discuss that range later). Those values are the only physics predictions, rather than just speculations. The LHC will run with higher luminosity after an upgrade, beginning in late 2021 if pandemic work stoppages do not delay it. That increases the possibility of discovery, though not very much. A higher energy collider is needed. From what we know now, a collider with twice the LHC energy range would probably suffice, and cover the region of gluino masses to about 5 GeV.

The concluding chapter of the 2017 edition tell us:

The compactified M-theory implies that three superpartners (and only three) will be observed at the LHC in the current three-year run (assuming the full integrated luminosity is achieved). These are the gluino and the charged and neutral winos.

Presumably he’s talking about the LHC Run II (2015-18) which did meet its luminosity goals, without any hint of the three superpartners. I don’t yet have access to the later parts of the 2021 edition to see what they say about this.

This isn’t the first time Kane has published multiple editions of ever changing “predictions” about supersymmetry. At one point I compared the 2000 and 2013 editions of “Supersymmetry and Beyond”, you can see the results here.

First some items on the mathematics side:

• The latest AMS Notices has some memorial pieces about Vaughan Jones and Robert Hermann. I contributed a piece to the Hermann memorial, for more about him, see here.
• If you read French you might enjoy Yves André’s Dix regards sur la mathématique contemporaine, freely available here.
• There’s a wonderful overview of various conjectures in number theory last year from Barry Mazur, About Main Conjectures.
• The Harvard Math department seems to not have had a lot of luck with its funders. Last spring they had to close their Program in Evolutionary Dynamics, which was funded by Jeffrey Epstein. The very active Center of Mathematical Sciences and Applications has been funded by the Evergrande Group, a real estate investment company that has now run into serious financial problems. I haven’t heard what the implications will be for the CMSA in the future.

On the physics side:

• The 2021 Physics Nobel Prize will be announced tomorrow morning. I gave up predicting these things after this prediction back in 2004.
• Gian Giudice has put on the arXiv a written version of his Theory closing talk at LCHP2021. He ends with
  

  These are interesting times for particle physics: times of great uncertainty, in which our physics perspective is changing, and in which we are laying the foundations for the future of our field. As a community, we must rise to the challenge.

  What worries me is that the much of the rest of the article contains a lot of

  1. Arguing for multiverse pseudo-science:
     

     The multiverse describes a physical reality that challenges the presumption that there must be a single unified theory in the deep UV. In a sense, it is the ultimate Copernican revolution since not even the patch of the universe we live in is special. It implies a revision of the cosmological principle because the universe is approximately homogeneous and isotropic only within our horizon, but may be globally highly non-homogeneous. The multiverse is not an abstract idea, but it is a generic consequence of a large class of inflationary theories, where unavoidable quantum fluctuations of the inflaton spark a chain process with eternal creation of regions that expand faster than the surrounding space.

     The multiverse is actually a familiar instrument of our everyday physics toolkit.

  2. Arguing against the fundamental significance of symmetry principles:
     

     There are also theoretical indications for questioning the concept of symmetry. It is nowbelieved (and to a certain extent proven) that any global symmetry is violated at the level of quantum gravity. This means that any global symmetry that we observe in nature is only an accidental effect of looking at a system without sufficient short-distance resolution. The case of gauge symmetries is more subtle. Gauge symmetries are not real physical symmetries, in the sense
     that they don’t correspond to an invariance under a physical transformation, but only to a redundancy of the coordinate parametrisation. We often confuse our students on this point by showing them the Mexican-hat potential and leading them to believe that there is a degeneracy of vacua, when in reality there is only one single vacuum state that breaks EW symmetry, as it is clear from the fact that the physical spectrum doesn’t contain any Goldstone boson corresponding to zero-energy excitations. Gauge symmetries may not be as fundamental as we thought, but only an emergent phenomenon. They could be a mirage of a different reality that takes place at a more fundamental level.

  It’s looking depressingly possible that leaders of the field will push through as new “foundations for the future of our field” the argument that “the multiverse did it and symmetry is a mirage.” Instead of moving forward, the field will take a huge step backwards.