News and Travels

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 But then I realized he had labeled these “Traditional Questions”, in Fraktur font to emphasize how old and out of date they were.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.
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This Week’s Hype

The latest from the BBC:

String theory – a simple way to understand the universe

Not worth more comment than it’s another reminder that this nonsense continues to be heavily promoted in our most prominent and respected mass media. I’m beginning to doubt we’re going to be rid of it in my (or anyone’s) lifetime.

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String Theory and the Real World

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.

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Various Math and Physics News

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.

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Visit to Brown

I spent yesterday up in Providence, visiting the Theoretical Physics Center at Brown, and giving a talk there (slides are available here, newer version of a paper here), At some point a recording of the talk should appear online. In the talk I tried to emphasize some basic things which it took me a very long time to appreciate:

  • The ways in which Euclidean QFT is very different than Minkowski space-time QFT, in particular the necessity of having a distinguished imaginary time vector, breaking SO(4) invariance, in order to recover Lorentz (SL(2,C)) invariance.
  • The way in which Minkowski space-time shows up when you do twistor theory in Euclidean space-time (see the pictures in the slides). This again makes clear the way SO(4) invariance is broken.

While I’m making a proposal for how to get gravity out of chiral 4d geometry, I’ve never been that expert in GR, and GR is the focus of much of the theory community these days, in particular the theorists at Brown. So, they had lots of questions about what the implications of this are for GR that I couldn’t answer. I’ll keep thinking more about this and may some day start to have answers (or maybe GR experts will find this proposal interesting enough to figure out the answers themselves).

I was invited to give the talk by Stephon Alexander, and got to spend some time talking with him while in Providence. He has worked in the past (see here) on ideas that bring together the gravitational and weak interactions in a similar way. More recently he has been working on ideas for how one might observe an unexpected chiral component to the gravitational interactions, and now has a grant from the Simons Foundation that will fund work in this area. Next week he’ll be here at Columbia giving an astronomy colloquium on the topic.

He also has a new book (his first was The Jazz of Physics) out, Fear of a Black Universe: an Outsider’s Guide to the Future of Physics. It’s quite interesting, with much of the earlier parts describing some of his experiences making a career for himself as a theorist, together with explanations of the physics background. The last part (in collaboration with Jaron Lanier) heads off in somewhat of a sci-fi direction, an excerpt is here.

A major theme of the book (with which I’m very sympathetic) is that the community doing this sort of theoretical physics desperately needs to get out of its current rut and open itself to new ideas, which often will come from “outsiders”. One aspect of being an “outsider” that Alexander has experienced is difference in racial background, but he’s concerned with a more general context of hostility to ideas that aren’t those currently favored by “insiders”. While he started out his career doing string theory, he has moved in different directions over the years. He explains that as a postdoc at SLAC he invited Lee Smolin to come and lecture on loop quantum gravity, something which was not at all well received by the local string theorists. While I’m quite interested for my own reasons to understand better what he has been doing with the physics of possible chiral effects in gravity, it was great to see his enthusiasm for and encouragement of ideas that don’t fit exactly into the narrow conception of the subject that now dominates all too much of the community doing fundamental theoretical physics.

Update: There’s video of the talk available here.

Posted in Book Reviews, Euclidean Twistor Unification | 7 Comments

Euclidean Twistor Unification, version 2.0

I’ve completely re-organized and largely rewritten my paper from earlier this year on Euclidean Spinors and Twistor Unification. Soon I’ll upload this as a revision to the arXiv, for now it’s available here. This new version starts from a very basic point of view about 4d geometry, leaving the technicalities about Euclidean QFT for spinors and the expository material about twistors to appendices.

Most ideas I’ve worked on over the years that seemed initially promising ultimately became more and more problematic the more I looked at them. This set of ideas keeps looking more and more solid. There are several (to me at least…) attractive aspects:

  • Spinors are tautological objects (a point in space-time is a space of Weyl spinors), rather than complicated objects that must be separately introduced in the usual geometrical formalism.
  • Analytic continuation between Minkowski and Euclidean space-time can be naturally performed, since twistor geometry provides their joint complexification.
  • Exactly the internal symmetries of the Standard Model occur.
  • The intricate transformation properties of a generation of Standard Model fermions correspond to a simple construction.
  • One gets a new chiral formulation of gravity, unified with the Standard Model.
  • Conformal symmetry is built into the picture in a fundamental way.

There’s more in this version about how quantum gravity fits into this, when formulated in terms of chiral variables (i.e. Ashtekar variables). This gives a new context for old questions about quantizing in these variables (this is in Eucldean signature, the other chirality is not space-time geometry but internal Yang-Mills geometry, and the imaginary time component of the vierbein is distinguished and given the dynamics of a Higgs field). I haven’t spent much time on this yet, but suspect this new context may help overcome problems that people trying to pursue quantum gravity in this chiral connection framework have run into in the past.

One common reaction I’ve gotten to these ideas is the one I myself had in the past: analytic continuation relates expectation values of field operators in Euclidean and Minkowski signature, so my left-handed SU(2) after analytic continuation gives part of Lorentz symmetry, not an internal symmetry. What took me a long time to realize is just how different Euclidean and Minkowski signature QFT is. Yes, Schwinger functions and Wightman functions can be related by analytic continuation (in a rather subtle way, the Wightman functions aren’t functions, but boundary values of holomorphic functions). But at the level of states and operators things are very different. It’s just not true that there is some holomorphic formulation of QFT states and operators, with Euclidean and Minkowski space restrictions related by analytic continuation. There’s a lot of explanation about this in the paper.

One objection I’ve run into is that by distinguishing a direction in Euclidean space I’m breaking Lorentz symmetry. What’s true is quite the opposite: having such a distinguished direction is needed to get Lorentz symmetry after analytic continuation. If you want to start in Euclidean space and get Lorentz symmetry, you have to do something like distinguish a direction and get an Osterwalder-Schrader reflection in that direction, which you need to get from SO(4) to SL(2,C). From the other direction, if you start in Minkowski space-time and analytically continue, you have a choice of lots of possible Euclidean slices to analytically continue to. You need to pick one, and that will distinguish an imaginary time direction. This is most easily seen in the twistor formalism, where the Minkowski space-time geometry is determined by a quadratic form that picks out a 5-dimensional hypersurface in PT. This will project down to an imaginary time = 0 subspace of Euclidean space-time, which picks out the imaginary time direction.

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This Week’s Hype

Today Quanta has One Lab’s Quest to Build Space-Time Out of Quantum Particles. No, this kind of experiment is not going to “Build Space-Time”, now or ever. This kind of obfuscation about quantum gravity advances neither fundamental physics nor the public understanding of it, quite the opposite. The article does make clear what the motivation is: deal with the problem that

String theory, still the leading candidate to replace the Standard Model, has often been accused of being untestable.

by claiming that it somehow can be tested in a lab.

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Conversations on Quantum Gravity

Things for many years now have been going badly for string theory on the public relations front. Today the Economist has Physics seeks the future: Bye, bye, little Susy, where one finds out that:

But, no Susy, no string theory. And, 13 years after the LHC opened, no sparticles have shown up. Even two as-yet-unexplained results announced earlier this year (one from the LHC and one from a smaller machine) offer no evidence directly supporting Susy. Many physicists thus worry they have been on a wild-goose chase…
Without Susy, string theory thus looks pretty-much dead as a theory of everything. Which, if true, clears the field for non-string theories of everything.

Unfortunately for the public understanding of science, this is followed by

But at the moment the bookies’ favourite for unifying relativity and the Standard Model is something called “entropic gravity”… in the past five years, Brian Swingle of Harvard University and Sean Carroll of the California Institute of Technology have begun building models of what Dr Verlinde’s ideas might mean in practice, using ideas from quantum information theory.

For something much more anecdotal, on Saturday night I was having dinner outside in a hut during a rainstorm on the Upper East Side (having fled an aborted Central Park concert), and started talking to a couple seated nearby. When informed I taught math and did physics, one of them recommended Carlo Rovelli’s new book to me, and said he hoped I wasn’t doing string theory. Luckily I could reassure him about that.

This morning I found out about Conversations on Quantum Gravity, a fascinating book published by Cambridge that appeared online today, hard copies for sale in November. It consists of interviews about quantum gravity put together by Dutch string theorist Jay Armas, starting in 2011. The scale of this project is immense: there are 37 interviews, most of them rather long and detailed, making up a book of 716 pages. What I’m writing here is based on a day’s worth skimming of the book. I’ll likely go back again and look more carefully at parts of it.

Roughly half the interviewees are string theorists, with the author making a concerted effort to also include non-string theory approaches to quantum gravity. I made the mistake of starting off by reading some of the string theorist interviews, which was rather depressing. By the end of the day, after making my way through about 20 long interviews with string theorists, with few exceptions the story they were telling was one I’m all too familiar with. It’s roughly

We don’t actually know what string theory is, just that it’s a “framework” that encompasses QFT and much more. We can’t predict anything with it now and don’t see any plausible way of predicting anything in the future, but the theory is a successful theory of quantum gravity, unlike our competition. There is no good reason for people to be working on anything else.

For example, here’s Cumrun Vafa:

If a young student asks you what approach to quantum gravity they should work on, what would your answer be?

There is no question that string theory is the right framework to understand quantum gravity. By this I mean that it is closer to the truth than any other existent theory.

Is it worth exploring other approaches?

Well . . . certainly being close-minded is not good. We should be open to other developments. But the fact that there exist other subjects does not justify exploring them if they are not on equal footing with string theory.

and here’s Edward Witten:

Due to the lack of experimental data, there exist a plethora of different approaches to quantising gravity. Which of these approaches, in your opinion, is closer to a true description of nature and why?

I would say your premise is a little misleading. String theory is the only idea about quantum gravity with any substance. One sign is that where critics have had interesting ideas (non-commutative geometry, black hole entropy, twistor theory) they have tended to be absorbed as part of string theory.

and David Gross:

So you don’t think that other approaches like loop quantum gravity have . . .

Loop quantum gravity is total BS. I mean, it’s really not worth discussing it. Don’t put that in the book. But, it really isn’t.

Luckily Armas doesn’t take up Gross on the suggestion that loop quantum gravity is not worth discussing, interviewing quite a few people who are working on research programs that have grown out of it. I got much more out of these interviews, which were very different in tone and content than the ones with string theorists. Many of them gave a very clear account of the technical problems these approaches have encountered, referring to very specific well-defined models and calculations. Instead of the triumphalist claims and vague speculation of the string theorists there was a careful explanation of exactly what they were trying to do and the problems they were trying to overcome.

There’s a huge amount worth reading in these interviews, perhaps I’ll later add some more pointers. A couple specific examples that occur to me right now are Steve Carlip’s careful discussion of the quantization of the toy model of 2+1 dimensional gravity, and Lee Smolin’s very personal account of his frustration at the reception of his book “The Trouble With Physics”.

If your institution is paying Cambridge for access, you should take advantage of this now and take a look. Congratulations to Jay Armas for bringing us this material.

Update: There’s a new preprint out by historian of science Sophie Ritson, Constraints and Divergent Assessments of Fertility in Non-empirical Physics in the History of the String Theory Controversy, which examines in detail the arguments of the string wars and later over how to evaluate string theory. While I don’t think there’s a single reference in the 716 page Armas book to anything I’ve written, my views do make an appearance in this article.

Update: There’s a linked editorial in the Economist Fundamental physics is humanity’s most extraordinary achievement, which (rather optimistically) sees the current state of affairs as:

Supersymmetry is a stalking horse for a yet-deeper idea, string theory, which posits that everything is ultimately made of infinitesimally small objects that are most easily conceptualised by those without the maths to understand them properly as taut, vibrating strings.

So sure were most physicists that these ideas would turn out to be true that they were prepared to move hubristically forward with their theorising without experimental backup—because, for the first decades of Supersymmetry’s existence, no machine powerful enough to test its predictions existed. But now, in the form of the Large Hadron Collider, near Geneva, one does. And hubris is turning rapidly to nemesis, for of the particles predicted by Supersymmetry there is no sign.

Suddenly, the subject looks wide open again. The Supersymmetricians have their tails between their legs as new theories of everything to fill the vacuum left by string theory’s implosion are coming in left, right and centre.

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Some Math Items

Some math items that may be of interest:

Update: The Scholze review has been removed (temporarily?). A cached version is here.

Update: The review was temporarily removed just because what was posted wasn’t a finalized version, this is explained here. They should repost once Scholze has a chance to make any final edits.

Update: The review is back up.

Update: Michael Harris has a new substack site, where he’ll be writing about the mechanization of mathematics. I’m glad to see someone doing this from his point of view.

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More of the Same (Physics, Math and Unification)

I was going to just provide the following links with a some comments, but decided it would be a good idea to put them into what seems to me the larger context of where we are in fundamental physics and its relationship to mathematics.

For the latest on the conventional physics approach to unification (GUTS, SUSY, strings, M-theory), there’s:

  • The Lex Fridman podcast has an interview with Cumrun Vafa. Going to the section (1:19:48) – Skepticism regarding string theory) where Vafa answers the skeptics, he has just one argument for string theory as a predictive theory: it predicts that the number of spacetime dimensions is between 1 and 11.
  • A second edition of Gordon Kane’s String Theory and the Real World has just appeared. One learns there (page 1-19) that

    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.

For the latest in mathematics and the interface of math and physics, there’s

About the first two links, I’m at a loss for words.

The second two are extremely interesting topics indicating a deep unity of number theory, geometry and physics. They’re also not topics easy to say much about in a blog posting. In the Fargues-Scholze case that’s partly because the new ideas they have come up with relating arithmetic and geometry are ones I don’t understand very well at all (although I hope to learn more about them in the future). The connections they have found between representation theory, arithmetic geometry, and geometric Langlands are very new and it will likely be quite a few years before they are well understood and their implications well-developed.

In the Gaiotto-Witten case, some of what they discuss is very familiar to me: geometric quantization has been a topic of fascination since my student days, and one major goal of my QM book was to work out in detail (for the case of $\mathbf R^{2d}$) some of the subtleties about quantization that they discuss. For co-adjoint orbits in Lie algebras, geometric quantization has a long history, and “brane quantization” may or may not have anything new to say about this. For moduli spaces of vector bundles on Riemann surfaces, and Hitchin moduli spaces of Higgs bundles on Riemann surfaces, “brane quantization” might come into its own.

There is a fairly short path now potentially connecting fundamental unifying ideas in number theory and geometry to our best fundamental theories in physics (and seminars on arithmetic geometry and QFT are now a thing). The Fargues-Scholze work relates arithmetic and the central objects in geometric Langlands involving categories of bundles over curves. These categories in turn are related (in work of Witten and collaborators) to 4d TQFTs based on twistings of N=4 super Yang-Mills. This sort of 4d QFT involves much the same ingredients as 4d QFTs describing the Standard Model and gravity. For some better indication of the relation of number theory to this sort of QFT, a good source is David Ben-Zvi’s lectures this past semester (see here and here). I’m hopeful that the ideas about twistors and QFT in Euclidean signature discussed here will provide a close connection of such 4d QFTs to the Standard Model and gravity (more to come on this topic in the near future).

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