Two New Quantum Field Theory Books

Posted on April 11, 2022 by woit

I’ve recently noticed that two very good new books on quantum field theory have become available, one aimed more at mathematicians, one purely for physicists.

What Is a Quantum Field Theory?

Available online now from Cambridge University Press (actual printed books to come soon) is mathematician Michel Talagrand’s What Is a Quantum Field Theory?. While it’s subtitled “A First Introduction for Mathematicians” and definitely aimed more at mathematicians than physicists, it’s a wonderful resource for anyone who wants to understand exactly what a quantum field theory is.

Like many mathematicians, Talagrand tried to learn about quantum field theory first from physics textbooks, which tend to avoid any precise definition of even the basics of the subject. He soon found what was the best source for someone looking for more precision, Gerald Folland’s 2008 Quantum Field Theory: A Tourist Guide for Mathematicians. Folland’s book is extremely good, but also extremely terse. In 325 pages it covers more carefully the material of an old-style QFT book such as Schweber’s 900 page or so An Introduction to Relativistic Quantum Field Theory from 1961. Talagrand is covering much the same material, but with 742 pages to work with he is able (unlike Folland) to work out many topics in full detail, providing something previously unavailable anywhere else.

Both Folland and Talagrand have written books with much the same goal: to as precisely as possible explain the details of the renormalized perturbative expansion of QED. There is little overlap with the work of mathematical physicists who have aimed at rigorous non-perturbative constructions of quantum field theories. They are using canonical quantization methods and don’t overlap much with many of the more recent physics QFT textbooks, which are based on path integral quantization and aimed at getting to non-abelian gauge theories and non-perturbative techniques as quickly as possible.

When I was learning QFT not that long after the advent of the Standard Model, I had little patience for fat QFT books about perturbative QED and canonical methods. Why not just write down the path integral and start calculating? Over the years I’ve realized that things are not so simple, with canonical quantization and operator fields giving a perspective complementary to that of the path integral. Among the more modern books, volume 1 of Weinberg’s three-volume series is the one that best gives this different perspective, and is most closely related to what Talagrand is covering.

For mathematicians, Talagrand’s book is a great place to start. For physicists, Weinberg’s is an important perspective to get to know. If you’re reading Weinberg and want more detail about precisely what is going on, Talagrand’s new book would be a very good place to turn for help.

Quantum Field Theory: An Integrated Approach

Over the years I’ve often consulted various parts of Eduardo Fradkin’s notes on quantum field theory on his web pages. On some basic topics I found these to give very clear explanations of things that were done in a confusing way elsewhere. After recently hearing that the notes are now a book from Princeton University Press, I ordered a copy, which recently arrived.

Fradkin’s book has not much overlap with the material in the Talagrand book described above, and is somewhat different than traditional high energy physics-oriented QFT books. It tries as much as possible to integrate the high energy physics point of view with that of condensed matter and statistical mechanics. Path integral methods are then fundamental. Unlike many other modern QFT textbooks that aim at getting to the details of perturbative Standard Model calculations, Fradkin is more oriented towards getting as quickly as possible to non-perturbative techniques and models of interest in statistical mechanics. He gives a good introduction to various of the modern non-perturbative QFT techniques that have been developed in recent decades, often motivated by the so far only partially successful attempt to come to terms with a strongly-interacting gauge theory like QCD.

While most of the book is quite good, the first few pages aren’t, and will immediately drive away mathematicians who might pick it up. The material in these pages about group theory uses bad terminology (for Fradkin, the “rank” of a Lie group is its dimension and the fundamental representation of SU(n) is the “spinor” representation) and sometimes is just completely wrong. On the second page of the first chapter after the introduction, he wants to explain why the Lorentz group is non-compact, in contrast to SO(3). To explain why SO(3) is compact he starts by mistakenly arguing that since it leaves the unit two-sphere invariant the points of SO(3) and of the unit two-sphere are in one-to-one correspondence, showing the volume of SO(3) is $4\pi^2$. This paragraph should be deleted in future editions of the book.

That this kind of thing can make it into a book like this is remarkable, but unfortunately relativistic QFT books and other sources (e.g. here) don’t always get right basic facts about the Lorentz and rotation groups. I once tried to do my part to remedy this, see here.

Update: John Collins has here an article that provides a careful discussion of scattering in QFT, starting with the basics, which could be thought of as part of a QFT book. This may be of interest to both physicists and mathematicians who want to see something less superficial than many text book discussions.

Posted in Book Reviews | 35 Comments

ABC on NHK

Posted on April 9, 2022 by woit

There will be a documentary broadcast tomorrow in Japan on Mochizuki’s claimed proof of the abc conjecture. I was interviewed for this by the filmmakers last year, but don’t know anything about whether and how that footage will be used. I’d be curious to hear reports from any Japanese-speaking readers who see the documentary tomorrow.

Over the years there has been a detailed coverage of this story here on the blog. To make it more accessible, I’ve added an abc conjecture category. In case the documentary doesn’t make this clear, the current consensus of experts in the field is that there is no proof. Peter Scholze and Jacob Stix identified a problem with Mochizuki’s proof in 2018 (discussed in detail by Scholze and others here), and Mochizuki has not provided a convincing answer to their objections. No one else (including the journal editors who published the proof in PRIMS) has been able to provide a clear explanation of the problematic part of the proof.

Update: NHK has two web pages summarizing the content of the program, see here and here for English translations.

Taylor Dupuy is still making implausible claims that Scholze’s criticism of the proof is invalid. To judge for yourself, see here a long detailed discussion of the issue between them involving several other experts.

Reports I’m seeing from those who have watched the program say that it does correctly explain that the proof is not accepted by many experts.

Posted in abc Conjecture | 16 Comments

The Anti-Science Movement

Posted on April 6, 2022 by woit

I noticed recently that Stony Brook is hosting next week a panel discussion devoted to

a conversation about one of the most grave challenges to confront humanity: the anti-science movement.

There is a truly grave challenge being referred to, but a serious mistake is being made about the nature of the challenge. In particular, there’s no evidence of an “anti-science” movement, quite the opposite. Across the globe, if you ask people what profession they respect the most, “scientist” comes out on top (see here). Likely the organizers have in mind climate denialists and anti-vaxxers as prime examples of “anti-science” behavior, but in my experience such people typically show a great devotion to pointing to scientists, scientific results and scientific papers to justify themselves. An example would be Lubos Motl, who has put out literally thousands of pages on his blog about climate and COVID science (by the way, his blog seems to have gone “by invitation only”, anyone know what that’s about?).

The problem isn’t “anti-science”, but bad science, promoted for ideological reasons. This is part of a larger truly grave challenge to humanity, that of our information environment being flooded with untruth, on a scale that dwarfs the output of the Ministry of Truth that Orwell foresaw. For years now we’ve been living with this in the form of phenomena like Trumpism, and the past few weeks have seen the Russian government exploiting these methods to conduct a campaign of brutal slaughter. I don’t know what the best way to address this challenge is, but unless something can be done, humanity has an ugly and disturbing future ahead of it.

Sticking to the problem of what to do about the promotion of bad science, there at least I have some experience trying to do something about one example of it (although with very limited success). This problem deserves attention and a panel discussion, but a panel in which four of six members have devoted a significant part of their careers to promoting a failed scientific research program is a really odd choice.

The underlying thorny issue is that of how to evaluate scientific claims. Given the complexities of controversial science, non-experts generally have little choice but to try and identify experts and trust what they say. A major societal role of elite institutions is to provide such experts, ensuring that they provide trustworthy expertise, untainted by ideology or self-interest. A large part of what is going on these days seems to me to reflect a loss of faith in elite institutions, with an increasing perception that these are dominated by a well-off class pursuing not truth, but their own interests. As a product of such institutions I’m well aware of both their strengths and their weaknesses. We need them to do better, and in this case Stony Brook should come up with a better panel.

Update: I’ve heard that Lubos himself shutdown the blog, unwilling to agree to follow rules Google was now enforcing.

Posted in Fake Physics, Uncategorized | 42 Comments

Is Space-Time Really Doomed?

Posted on March 31, 2022 by woit

For many years now the consensus in a dominant part of the theoretical physics community has been that the center of attention should be on the problem of quantizing gravity, and that conventional notions of quantum theory and space-time geometry need to be abandoned in favor of something radically different. The slogan version of this is “Space-Time is Doomed.”

Ever since my student days long ago, I’ve spent a lot of time looking into the problems of quantum gravity and what people have tried to do to address these problems. The highly publicized attempts to get known physics out of radically different degrees of freedom that I’ve seen haven’t seemed to be making any progress, remaining very far from anything like known physics. In the case of string theory, which also claimed to be able to get particle physics, there was at one point a (highly over-hyped) relatively well-defined proposal that one could discuss, but that’s no longer the case.

Recently things have changed as I’ve become convinced of the promise of certain specific ideas about four-dimensional geometry involving twistors and Euclidean space-time signature. I’ve written about these here and on the blog, and have given some talks (see here and here). These ideas remain speculative and incomplete, but I think they provide some new ways of thinking about the problems of quantizing gravity and unifying it with the other forces.

The existence of a yearly essay competition gave me an excuse to write something about this which I just finished yesterday and sent in, with the title Is Space-Time Really Doomed?. After spending some time on a diversion into arithmetic geometry, I’ve been getting back to seriously thinking about this topic, looking forward to having time in coming months to concentrate on this. I hope the essay will encourage others to not give up on 4d geometry as doomed and unquantizable, but to realize that much is there still waiting to be explored.

Update: The essay is now on the arXiv here.

Update: Awards for this announced here. I got an honorable mention.

Posted in Euclidean Twistor Unification | 22 Comments

2022 Abel Prize to Dennis Sullivan

Posted on March 23, 2022 by woit

This year’s Abel Prize has gone to topologist Dennis Sullivan, for the announcement see here, with more information about Sullivan and his work here. There are press stories at Nature, the New York Times, Quanta, and elsewhere.

Sullivan was one of the leading figures in great advances in understanding the topology of manifolds in higher dimensions during the late 60s and 70s. Some of the best of his early work for many years was only available if you could find a copy of unpublished mimeographed notes from a 1970 MIT course. In 2005 a Tex’ed version of the notes was finally published (available here). This includes as a postscript Sullivan’s own description of this work, how it came about, and how it influenced his later work.

This was followed by wonderful work on rational homotopy theory, making use of differential forms. For this, see Sullivan’s 1977 Infinitesimal computations in topology, and lecture notes on this by Phil Griffiths and John Morgan. In later years Sullivan’s attention turned to subjects with which I’m not very familiar: topics in dynamical systems and the development of what he called “string topology”.

Since 1981 Sullivan has held the Einstein chair at the CUNY Graduate Center here in NYC, running a seminar each week that concentrates on the relation between topology and QFT. For many years these were held in a Russian style, going on for multiple hours, possibly with a break, until all participants were exhausted. There’s a remarkable collection of videos of these lectures at the seminar site, including many going way back into the 80s and 90s, with video recorded at a time when this was quite unusual (more recent ones are on Youtube).

When I first came to Columbia Sullivan was often here attending and giving lectures, for many years splitting his time between Paris (where he held a position at the IHES), New York and Rio. The Abel Prize biography explains

In 1981, Sullivan was made the Albert Einstein Chair in Science (Mathematics) at the Graduate School and University Center of The City University of New York. He kept his position at IHES and spent the next decade and a half shuttling between Paris and New York, often on Concorde.

Some of the various stories I heard about Sullivan’s lifestyle at the time involved his having multiple apartments in New York, which he used to host a variety of visiting mathematicians. Another story I heard directly from him was about how he survived an attempted car-jacking in Brazil, during which he was shot, but managed to escape and drive himself to a hospital for treatment. I had first heard about this from Mike Hopkins several years before. When I asked Hopkins why he had become a topologist, he said that one factor was the inspiring example of people like Sullivan who worked in the field, jokingly characterizing it as involving “real men who got into gun-fights”.

In 1997 Sullivan traded the IHES position for one at Stony Brook, and over the years has unfortunately been seen less often here at Columbia. Congratulations to him on the well-deserved prize!

Posted in Uncategorized | Comments Off on 2022 Abel Prize to Dennis Sullivan

A Few Items

Posted on March 4, 2022 by woit

Some short items and links:

  • Among possible futures that I never would have dreamed of during my student days was that someday my Nobel-prize-winning undergraduate advisor would
    “try to rile” my Nobel-prize-winning graduate school professor at a Bohr Centennial celebration by quoting me. I hope the quote at least was one I would agree with.
  • Also on the topic of hoping I agree with what I say publicly, there’s an NHK documentary about Mochizuki and the abc conjecture that has recently been finished, was supposed to air in Japan this weekend, now delayed til next month due to more timely news from Ukraine. I did an interview with the filmmakers here in NYC last year and they talked to many other people. No idea how they’ll manage to deal with this controversial story, coming from a Japanese perspective.
  • At Quanta magazine, another article about the “naturalness problem”, headlined A Deepening Crisis Forces Physicists to Rethink Structure of Nature’s Laws. This has the usual problem with such stories of assigning to the Standard Model something which is not a problem for it, but only for certain kinds of speculative attempts to go beyond it. John Baez makes this point in this tweet:

    Indeed, calling it a “crisis” is odd. Nothing that we really know about physics has become false. The only thing that can come crashing down is a tower of speculations that have become conventional wisdom.

    James Wells has a series of tweets here, starting off with

    The incredibly successful Standard Model does not have a Naturalness problem. And if by your criteria it does, then I can be sure your definition of Naturalness is useless.

    He points to a more detailed explanation of the issue in section 4 of this paper.

  • My criticisms of some Quanta articles are motivated partly by the fact that the quality of the science coverage there is matched by very few other places. If you want to work there, they have a job open.
  • But if you really want to cash in on gazillionaire money going into mathematics, you might want to try for some of the $20 million that crypto entrepreneur Charles Hoskinson is giving Carnegie Mellon to establish the Hoskinson Center for Formal Mathematics. Early in his career Hoskinson was in a Ph.D. program in analytic number theory, but bailed and later joined Ron Paul’s presidential campaign, and ended up in crypto since “When Bitcoin came out, it was like the spiritual successor to what Ron Paul was talking about” (see here).
  • Someone who is not going to be getting Hoskinson funding is Michael Harris, whose The Silicon Reckoner you should follow for an alternate take on “formal mathematics”. For the reaction to such criticism from the formalizers, you can check out this Zulip Chat archive, and then responses from Harris here.
  • For Grothendieck news, see here, here and here.

Update: There’s a statement out today from Breakthrough Prize Laureates strongly criticizing Russia’s invasion of Ukraine. There’s also a truly appalling statement from the Breakthrough Prize Foundation itself, not in the least critical of Russia or Putin and abusing the memory of Stephen Hawking. Witten characterizes the Foundation statement as “disappointingly vapid”.

Update: Milner seems to have realized that refusing to criticize Putin was not a tenable position. A new statement from the Breakthrough Prize Foundation starts off with:

As the terrible war in Ukraine continues, with casualties and atrocities mounting, the Breakthrough Prize Foundation strongly condemns Russia’s invasion of Ukraine and its unprovoked and brutal assaults against the civilian population.

and announces a further \$ 3 million donation:

the Foundation today pledges a further \$3 million in funding to support physicists, life scientists and mathematicians who have been forced to flee from Ukraine. We hope that this donation will help talented researchers contribute to human knowledge in such dark times.

The Breakthrough Prize Foundation stands together with the Ukrainian people, its scientists and their families.

Posted in Uncategorized | 32 Comments

ICM 2022 and the Invasion of Ukraine

Posted on February 21, 2022 by woit

The news this evening that Russia is sending troops into the Eastern Ukraine and in effect announcing annexation of at least part of the Ukraine carries extremely disturbing implications for the whole world. On a much more minor scale of importance, I don’t see how the IMU has any choice but to cancel this year’s ICM planned for St. Petersburg in July.

Four years ago when the IMU chose St. Petersburg over Paris for the 2022 ICM I commented here on this blog:

It does seem to me though that in these worrisome times, when offered the choice between the world’s most active opponent of liberal democracy and one of the great remaining healthy liberal democracies, the other choice than the one the IMU made would have been the better one…

I agree that in general it’s best to keep mathematics and the ICM out of politics. A question to think about though for those who know the history of the 1930s is that of whether there was some point during the rise of Fascism that one would stop thinking it was a good idea to have the ICM in a Fascist capital. We’re not yet far along the horrific path of the 1930s, but maybe that just means that all should be thinking about what can be done to keep the world from going down that path again.

I sympathize with many who felt that the decision to hold the ICM in Russia was an important way to support Russian mathematicians and a reasonable gamble that Putin would not take his country down the path he now appears to have chosen. But right now it’s looking like that gamble failed and the IMU will have to figure out what to do about its mistake.

I don’t want to host a general political discussion here, especially not with the all too many people I’ve heard from who don’t have a problem with burying liberal democracy. If your comment is not about the ICM, please don’t submit it.


Update: On February 10 an organization of Ukrainian mathematicians emailed the ICM invited speakers asking them to cancel their talks (and on January 31 had emailed the AMS leadership). I’m curious to know if any responded to this, and if the Russian military invasion will lead to some decisions to cancel talks.


Update: According to this from @UkrainianMath, the AMS position was recently that “the AMS leadership is closely tracking the situation surrounding the ICM and believes that it is still premature to advocate a boycott”. The “premature” indicates that there is some point at which AMS leadership agrees that the ICM should be canceled. Is the situation of Russian troops occupying the Eastern Ukraine still “premature”, or will the AMS wait for them to take Kiev?

Update: I noticed there’s an AMS-NSF-Simons-ICM Travel Grant program to fund ICM participation by US mathematicians. It was supposed to announce awards this month. Will this program go forward or will the grants be canceled?

Update: Many Russian mathematicians likely feel the same way about this as Edward Frenkel who calls this “a catastrophe for Russian people and all Slavic people”.

Update: The AMS has issued a statement urging the IMU to cancel the ICM and has suspended the AMS-NSF-Simons-ICM travel grant program

Update: There’s a statement out signed by invited ICM speakers. Unfortunately it has been overtaken by events, with little more than a request that the IMU “elaborate and announce contingency plans” in case of war, something that would have made sense a month ago, but not now. Nothing yet that I’m aware of from the IMU or other mathematical societies than the AMS.

Update: The French SMF has a statement calling on the IMU to not hold the ICM in Russia during 2022. In another statement, the London Mathematical Society “strongly recommends that the IMU not hold the ICM in Russia in July 2022.” Also in France, INSMI at the CNRS has this.

Update: There’s a long twitter thread about this here. It includes a contribution from Ian Agol: “As a chair of the topology selection committee, I requested @ICM2022
that the opening ceremony not be presided by a head of state (presumably Putin), but they were not willing to consider this.”
This makes clear the fundamental problem with deciding to hold the ICM in a country ruled by a fascist dictatorship. If you do this, you end up putting the conference under the control of the dictator, because anyone inside the country cannot oppose them. Those outside the country end up having to either go along with the dictator, or cancel the conference, and this is where the IMU is now.

Update: The IMU has issued a (rather empty) statement, saying that “The Executive Committee of the IMU is now assessing the situation”

Update: More statements from national math societies: Italy, Canada, Poland, Lithuania.

Updates: The IMU this morning has on their website:

The IMU Executive Committee is currently assessing the highly disturbing events that are taking place in Ukraine and their implications for the IMU. We will return with a statement as soon as it is available.

There are new statements calling for canceling the ICM from the European Mathematical Society, the Australian Mathematical Society, the Swedish Mathematical Society and others.

Update: Via @UkrainianMath, the latest from the IMU Secretary:

The IMU EC has been sitting in meetings for two days now, discussing the situation and how to respond properly. I ask for your understanding that it is more difficult for a global organization to meet and discuss this issue.

Of particular concern to us is how to find a possible way to carry out a General Assembly and an ICM if possible, but outside Russia. Furthermore, we do not want to cause damage to our Russian colleagues, who have spent endless hours preparing for an ICM.

The IMU webpage does contain a statement that we are working on this, and until we have reached a decision, which will be very soon, this is the best we can do.

Update: The IMU has announced that the ICM will take place as scheduled, but as a free fully virtual conference, not in-person in St. Petersburg. The IMU General Assembly will take place in-person, but at a different location still to be chosen, outside Russia.

Update: The ICM website (icm2022.org) no longer exists, with that address redirected to the IMU site mathunion.org. There had been no activity on the @ICM2022 Twitter account since Feb. 11, but now there’s a statement from four of the Russian mathematicians who had been involved in organizing the ICM:

We condemn the madness, the injustice, and the irreversibility of war that threatens the very existence of humanity. While our losses cannot be compared to the losses and the suffering of millions of people in the Ukraine, we are devastated to see all of our dreams and all of our work of many years ruined. The goals towards which we worked could not have been further from the horror that is happening and those responsible for it. Still, amid the ruins of our dreams, we feel left with an insurmountable debt that may take much longer than the life of our generation to be forgiven.

D. Belyaev, A. Okounkov, J. Pevtsova, S. Smirnov

Update: See here for a letter to the IMU from some mathematicians arguing against the decision to hold the ICM online.

Posted in Uncategorized | 51 Comments

Seminar talk on Euclidean Twistor Unification and the Twistor P1

Posted on February 14, 2022 by woit

Today I gave a talk via Zoom at the Algebra, Particles and Quantum Theory seminar series organized by Nichol Furey. The slides from the talk are here (I gather the talk was recorded and video might be available at some point).

This talk emphasized explaining the twistor geometry, integrating some of what I’ve learned over the last few months thinking about the “twistor $P^1$” (see here). For instance, one way to think of the basic object of Euclidean twistor theory is as $\mathbf {CP}^3$, together with a different real structure (the twistor real structure) than the usual one given by conjugation of complex coordinates. One thing that struck me while writing up these slides is that the Euclidean twistor story gets a lot of mileage out of identifying $\mathbf C^2$ and $\mathbf H$, together with taking as fundamental $\mathbf H^2$. It has always seemed possible that the octonions might have a role to play here; one way into that might be to think about identifying $\mathbf H^2$ with $\mathbf O$ in some analogous way to the $\mathbf C,\mathbf H$ story.

There’s nothing new here about any of the many open questions of how to use this geometrical framework to get a fully worked out dynamics that would include the Standard Model and gravity. After a detour into number theory and hyper-Kähler geometry for several months, I’m now getting back to thinking about those questions.

Update: Video of the talk is now available here.

Posted in Euclidean Twistor Unification | 17 Comments

This Way to the Universe

Posted on February 10, 2022 by woit

There’s a new popular book out this week by string theorist Michael Dine, This Way to the Universe, as well a a new Sean Carroll podcast interviewing him about the book and the state of particle theory research. According to Carroll, Dine represents the “insider view” of what is really going on in fundamental physics:

you’re getting what basically is the closest to a consensus view of what state particle theory and fundamental physics is right now.

Much of the book is a very conventional and straightforward attempt to explain modern particle theory/GR/cosmology to a general audience, featuring some explanations from Dine about specific attempts to go beyond the Standard Model that he has worked on. The last quarter or so of the book is about string theory and the multiverse. One odd thing about this is that the jacket copy is fraudulent, stating:

People assume string theory can never be tested, but Dine intrepidly explores how the theory might be investigated experimentally.

whereas there’s nothing like that in the book that I could find. Instead, on page 253 one finds about string theory

it’s not clear it’s right, or that it even makes definite predictions at all…
Many readers will know that string theory has been a lightning rod for criticism. In this chapter, we’ll understand why, on the one hand, the subject is so seductive, and on the other, its critics may have a point.

On page 295 there’s

In fact, the existence of states in string theory that really look similar to what we see around us is highly conjectural. This hasn’t stopped me nor my colleagues from writing many papers speculating on a stringy reality. Unfortunately, none of these papers can be said to be making a prediction from string theory. Typically the author likes one particular solution of string theory or another, and selects one feature that is distinctive and goes beyond the Standard Model. But apart from the arbitrariness of this choice, the author has closed his or her eyes to two huge problems with their proposal [the cosmological constant and, perhaps, supersymmetry breaking].

On the Carroll podcast, there’s this exchange (starting around 1:25):

MD: There are people who work actively trying to… On what they would call string phenomenology, but I think that at the moment, this is a hard topic and we just don’t understand well enough how in detail the theory could be related to nature… …strings are rather simple things, but the steps from there to things that look like the standard model, that look like general relativity, are pretty elaborate, and along the way, there are steps we don’t really understand…

SC: … Let me ask you how you respond to sort of the hardcore critics who might say something like this: In the 1980s, the first superstring revolution, people are going around saying like, yeah, we’re going to unify everything, we’re going to predict the mass of the electron and everything is going to be finished in 10 years. Then not only has string theory not made any predictions that you can test in an accelerator, but once we have the landscape of string theory, we’re saying that string theory is compatible with almost any set of particle physics you can have, and at that point, shouldn’t you just give up and move on to something else? It’s not a thing that’s going to give you any testable predictions at any point in the future.

MD: Well, I would basically say that that, all that is fair, but at some gut level, I don’t exactly agree. So first of all, I would say that in 1985, already, in this era of the first superstring revolution, Nathan Seiberg and I pointed out what has come to be known as the Dine-Seiberg problem, a very basic and fundamental obstacle to relating string theory to nature. And people have proposed possible solutions, some of which are interesting, but really, there’s… In the subsequent nearly 40 years, people have not put forward. So I’m on safe ground, I sort of took both sides of this issue.

SC: .. why not just give up if we think that string theory could predict anything at all given the landscape problem?

MD: Well, I think my own attitude is to sit somewhere on the fence, not to devote huge amounts of energy to it, but to allow string theory to inform my thinking about various kinds of issues.

Dine is referring here to the “Dine-Seiberg problem”, described in a 1985 paper with abstract

We argue that if the superstring is to describe our world, it is probably strongly coupled. Several other (unlikely) possibilities are discussed.

The paper is specifically about the effective potential for the dilaton, but the problem is generic and fundamental: you can’t get anything like the real world out of the perturbative superstring and you don’t know what strongly coupled string theory is (one can argue that AdS/CFT tells you what strongly coupled string theory is, but again, that looks nothing like the real world).

The really odd thing about Dine’s current comments about testability (besides that they contradict the jacket of his book) is that 15-20 years ago he was for a while one of the theorists most prominently making the case that string theory and the landscape could be tested. I wrote about this often here on the blog, see for instance here and here. The second of these postings is about a 2007 Physics Today article by Dine entitled
String theory in the era of the Large Hadron Collider that claimed:

A few years ago, there seemed little hope that string theory could make definitive statements about the physics of the LHC. The development of the landscape has radically altered that situation. An optimist can hope that theorists will soon understand enough about the landscape and its statistics to say that supersymmetry or large extra dimensions or technicolor will emerge as a prediction and to specify some detailed features.

The Physics Today piece was rather explicitly an answer to my criticisms of the string theory landscape as untestable pseudo-science, with a subtitle

The relationship between string theory and particle experiment is more complex than the caricature presented in the popular press and weblogs.

Fifteen years later, in this new book Dine says nothing about his earlier claims about testing string theory at the LHC, or that others clearly pointed out at the time what was wrong with them. He ends with this summary of the situation:

… one can adopt the landscape viewpoint, but then one has to acknowledge that, at this point in time, we have nothing like a complete theoretical framework in which to make any scientific investigation, and that there are facts hard to reconcile with this viewpoint. I, for one, find this quite unsettling.

His experience of the last fifteen years does not seem to have made him think any more charitably of string theory critics, who get this sneering description on page 269-70:

[they] view the subject with total disdain, often wearing their ignorance of even its most rudimentary aspects as a badge of honor.

One telling mistake in the book is its reference (page 117) to the “Clay Mathematics Institute of Peterborough, New Hampshire”, an indication of the all too typical theoretical physicist’s lack of knowledge of anything about mathematicians and the mathematics community.

Update: There’s an interesting conversation between Dine and Lisa Randall about the book and these issues, see here.

Posted in Book Reviews, Multiverse Mania | 9 Comments

Notes on the Twistor P1

Posted on January 29, 2022 by woit

I’ve just finished writing up some notes on what the twistor $P^1$ is and the various ways it shows up in mathematics.  The notes are available here, and may or may not get expanded at some point.  The rest of the blog posting will give some background about this.

One of the major themes of modern mathematics has been the bringing together of geometry and number theory as arithmetic geometry, together with further unification with representation theory in the Langlands program. I’ve always been fascinated by the relations between these subjects and fundamental physics, with quantum theory closely related to representation theory, and gauge theory based on the geometry of bundles and connections that also features prominently in this story.

The Langlands program comes in global and local versions, with the local versions at each point in principle fitting together in the global version. In the simplest arithmetic context, the points are the prime numbers p, together with an “infinite prime”. A major development of the past few years has been the recent proof by Fargues and Scholze that the arithmetic local Langlands conjecture at a point can be formulated in terms of the geometric Langlands conjecture on the Fargues-Fontaine curve.

Back in 2015 Laurent Fargues gave a talk at Columbia on “p-adic twistors”. I attended the talk, and wrote about it here, but didn’t understand much of it. The appearance of “twistors” was intriguing, although they didn’t seem to have much to do with Penrose’s twistor geometry that had always fascinated me. What I did get from the Fargues talk was that the analog at the infinite prime of the Fargues-Fontaine curve (which I couldn’t understand) was something called the twistor $P^1$, which I could understand. The relation to the Langlands program was a mystery to me. Some years later I did talk about this a little with David Ben-Zvi, who explained to me that his work with David Nadler (see for instance here) relating geometric Langlands with the representation theory of real Lie groups involved a similar relation between local Langlands at the infinite prime and geometric Langlands on the twistor $P^1$.

Over the past couple years I’ve gotten much more deeply involved in twistor theory, working on some ideas about how to get unification out of the Euclidean version of it. I’ve also been fascinated by the Fargues-Scholze work, while understanding very little of it. Back in October Peter Scholze wrote to me to tell me he had taken a look at my Brown lecture and was interested in twistors, due to the fact that the twistor $P^1$ was the infinite prime analog of the Fargues-Fontaine curve. He remarked that it’s rather mysterious why the twistor $P^1$ is what is showing up here as the geometrical object governing what is happening at the infinite prime. I was very forcefully struck by seeing that this object was exactly the same object that describes a space-time point in twistor theory and I mentioned this at the end of my talk in Paris back in late October.

Scholze’s comments inspired me to take a much closer look at the twistor $P^1$, beginning by trying to understand a bunch of things that were somehow related, but that I had never really understood. These ranged from Carlos Simpson’s approach to Hodge theory via the twistor $P^1$ to some basic facts about local class field theory, where one gets a simple analog for each prime p of the twistor $P^1$ and the quaternions. Along the way, I finally much better understood something else in number theory that had always fascinated me, see the story explained very sketchily in section 6.2. That the quantum mechanical formalism for a four-dimensional configuration space beautifully generalizes to all primes, with the global picture including an explanation of quadratic reciprocity is not something I’ve seen elsewhere in attempts to bring p-adic numbers into physics. I’d be very curious to hear if someone else knows of somewhere this has been discussed.

Anyway, these new notes are partly for my own benefit, to put what I’ve understood in one place, but I hope others will find something interesting in them. Now I want to get back to thinking about the open questions raised by the twistor unification ideas that I was working on before the last few months. A big question there is to understand what twistor unification might have to do with Witten’s ideas relating geometric Langlands with 4d QFT. Perhaps something I’ve learned by writing these notes will be helpful in that context.

Update: I’ve posted the notes, with an added abstract and a final section of speculations, to the arXiv, see here.

Posted in Euclidean Twistor Unification, Langlands | 8 Comments