*The following makes no claims to originality or any physical significance on its own. For a better explanation of some of the math and the physical significance of the use of quaternions here, see this lecture by John Baez. *

*I’ve been spending a lot of time thinking about spinors and vectors in four dimensions, where I do think there is some important physical significance to the kind of issue discussed here. See chapter 10 here for something about four dimensions. A project for the rest of the semester is to write a lot more about this four-dimensional story.*

Until recently I was very fond of the following argument: in three dimensions the relation between spinors and vectors is very simple, with spinors the more fundamental objects. If one uses the double cover $SU(2)=Spin(3)$ of the rotation group $SO(3)$, the spinor (S) and vector (V) representations satisfy

$$ S\otimes S = \mathbf 1 \oplus V$$

which is just the fact well-known to physicists that if you take the tensor product of two spinor representations, you get a scalar and a vector. The spinors are more fundamental, since you can construct $V$ using $S$, but not the other way around.

I still think spinor geometry is more fundamental than geometry based on vectors. But it’s become increasingly clear to me that there is something quite subtle going on here. The spinor representation is on $S=\mathbf C^2$, but one wants the vector representation to be on $V_{\mathbf R}=\mathbf R^3$, not on its complexification $V=\mathbf C^3$, which is what one gets by taking the tensor product of spinors.

To get a $V_{\mathbf R}$ from $V$, one needs an extra piece of structure: a real conjugation on $V$. This is a map

$$\sigma:V\rightarrow V$$

which

- commutes with the $SU(2)$ action
- is antilinear

$$ \sigma(\lambda v)=\overline \lambda v$$ - satisfies $\sigma^2=\mathbf 1$

$V_{\mathbf R}$ is then the conjugation-invariant subset of $V$.

If we were interested not in usual 3d Euclidean geometry and $Spin(3)$, but in the geometry of $\mathbf R^3$ with an inner product of $(2,1)$ signature, then the rotation group would be the time-orientation preserving subgroup $SO^+(2,1)\subset SO(2,1)$, with double cover $SL(2,\mathbf R)$. In this case the usual complex conjugations on $\mathbf C^2$ and $\mathbf C^3$ provide real conjugation maps that pick out real spinor ($S_{\mathbf R}=\mathbf R^2\subset S$) and vector

$(V_{\mathbf R}=\mathbf R^3\subset V=S\otimes S)$ representations.

For the case of Euclidean geometry and $Spin(3)$, there is no possible real conjugation map $\sigma$ on $S$, and while there is a real conjugation map on $V$, it is not complex conjugation. To better understand what is going on, one can introduce the quaternions $\mathbf H$, and understand the spin representation in terms of them. The spin group $Spin(3)=SU(2)$ is the group $Sp(1)$ of unit-length quaternions and the spin representation on $S=\mathbf H$ is just the action on $s\in S$ of a unit quaternion $q$ by left multiplication

$$s\rightarrow qs$$

(we could instead define things using right multiplication).

There is an action of $\mathbf H$ on $S$ commuting with the spin representation, the right action on $S$ by elements $x\in \mathbf H$ according to

$$s\rightarrow s\overline{q}$$

(this is a right action since $\overline {q_1q_2}=\overline q_2\ \overline q_1$).

This quaternionic version of the spin representation is a complex representation of the spin group, since the right action by the quaternion $\mathbf i$ provides a complex structure on $S=\mathbf H$. While there are no real conjugation maps $\sigma$ on the spin representation $S$, there is instead a quaternionic conjugation map, meaning an anti-linear map $\tau$ commuting with the spin representation and satisfying $\tau^2=-\mathbf 1$. An example is given by right multiplication by $\mathbf j$

$$\tau (q)=q\mathbf j$$

Note that in the above we could have replaced $\mathbf i$ by any unit-length purely imaginary quaternion and $\mathbf j$ by any other unit-length purely imaginary quaternion anticommuting with the first.

In general, a representation of a group $G$ on a complex vector space $V$ is called

- A real representation if there is a real conjugation $\sigma$. In this case the group acts on the $\sigma$-invariant subspace $V_\mathbf R\subset V$ and $V$ is the complexification of $V_\mathbf R$.
- A quaternionic representation if there is a quaternionic conjugation $\tau$. In this case $\tau$ makes $V$ a quaternionic vector space, in a way that commutes with the group action.

Returning to our original situation of the relation $S\otimes S= 1 \oplus V$ between complex representations, $S$ is a quaternionic representation, with a quaternionic conjugation $\tau$. Applying $\tau$ to both terms of the tensor product the minus signs cancel and one gets a real conjugation $\sigma$ on $V$.

What’s a bit mysterious is not the above, but the fact that when we do quantum mechanics, we have to work with complex numbers, not quaternions. We then have to find a consistent way to replace quaternions by complex two by two matrices when they are rotations and and complex column vectors when they are spinors (so $S=\mathbf C^2$ rather than $\mathbf H$).

In my book on QM and representation theory I use a standard sort of choice that identifies $\mathbf i,\mathbf j,\mathbf k$ with corresponding Pauli matrices (up to a factor of $i$):

$$1\leftrightarrow \mathbf 1=\begin{pmatrix}1&0\\0&1\end{pmatrix},\ \ \mathbf i\leftrightarrow -i\sigma_1=\begin{pmatrix}0&-i\\ -i&0\end{pmatrix},\ \ \mathbf j\leftrightarrow -i\sigma_2=\begin{pmatrix}0&-1\\ 1&0\end{pmatrix}$$

$$\mathbf k\leftrightarrow -i\sigma_3=\begin{pmatrix}-i&0\\ 0&i\end{pmatrix}$$

or equivalently identifies

$$q=q_0 +q_1\mathbf i +q_2\mathbf j + q_3\mathbf k \leftrightarrow \begin{pmatrix}q_0-iq_3&-q_2-iq_1\\q_2-iq_1 &q_0 +iq_3\end{pmatrix}$$

Note that this particular choice incorporates the physicist’s traditional convention distinguishing the $3$-direction as the one for which the spin matrix is diagonalized.

The subtle problem here is the same one discussed above. Just as the vector representation is complex with a non-obvious real conjugation, here complex matrices give not $\mathbf H$ but its complexification

$$M(2,\mathbf C)=\mathbf H\otimes_{\mathbf R}\mathbf C$$

The real conjugation is not complex conjugation, but the non-obvious map

$$\sigma (\begin{pmatrix}\alpha&\beta \\ \gamma & \delta\end{pmatrix})= \begin{pmatrix}\overline\delta &-\overline\gamma \\ -\overline\beta & \overline\alpha \end{pmatrix}$$

Among mathematicians (see for example Keith Conrad’s Quaternion Algebras), a standard way to consistently identify $\mathbf H$ with a subset of complex matrices as well as with $\mathbf C^2$, (giving the spinor representation) is the following:

- Identify $\mathbf C\subset \mathbf H$ as

$$z=x+iy\in \mathbf C \leftrightarrow x+\mathbf i y \in \mathbf H$$ - Identify $\mathbf H$ as a complex vector space with $\mathbf C^2$ by

$$q=z +\mathbf j w \leftrightarrow \begin{pmatrix}z\\ w\end{pmatrix}$$

Note that one needs to be careful about the order of multiplication when writing quaternions this way (where multiplication by a complex number is on the right), since

$$z+w\mathbf j= z+\mathbf j\overline w$$ - Identify $\mathbf H$ as a subset of $M(2,\mathbf C)$ by

$$q=z +\mathbf jw \leftrightarrow \begin{pmatrix}z&-\overline{w}\\ w& \overline z\end{pmatrix}$$

This is determined by requiring that multiplication of quaternions in the spinor story correspond correctly to multiplication of an element of $\mathbf C^2$ by a matrix.

With this identification

$$\mathbf i\leftrightarrow \begin{pmatrix}i&0\\ 0&-i\end{pmatrix},\ \ \mathbf j\leftrightarrow \begin{pmatrix}0&-1\\ 1&0\end{pmatrix},\ \ \mathbf k\leftrightarrow \begin{pmatrix} 0&-i\\ -i&0\end{pmatrix}$$

This is a bit different than the Pauli matrix version above, but shares the same real conjugation map identifying $\mathbf H$ as a subset of $M(2,\mathbf C)$.

There’s been very little blogging here the past month or so. For part of the time I was on vacation, but another reason is that there just hasn’t been very much to write about. Today I thought I’d start looking at the talks from this week’s Strings 2024 conference.

The weird thing about this version of Strings 20XX is that it’s a complete reversal of the trend of recent years to have few if any talks about strings at the Strings conference. I started off looking at the first talk, which was about something never talked about at these conferences in recent years: how to compactify string theory and get real world physics. It starts off with some amusing self-awareness, noting that this subject was several years old (and not going anywhere…) before the speaker was even born. It rapidly though becomes unfunny and depressing, with slides and slides full of endless complicated constructions, with no mention of the fact that these don’t look anything like the real world, recalling Nima Arkani Hamed’s recent quote:

“String theory is spectacular. Many string theorists are wonderful. But the track record for qualitatively correct statements about the universe is really garbage”

The next day started off with Maldacena on the BFSS conjecture. This was a perfectly nice talk about an idea from 25-30 years ago about what M-theory might be that never worked out as hoped.

Coming up tomorrow is Jared Kaplan explaining:

why it’s plausible that AI systems will be better than humans at theoretical physics research by the end of the decade.

I’m generally of the opinion that AI won’t be able to do really creative work in a subject like this, but have to agree that likely it will soon be able to do the kind of thing the Strings 2024 speakers are talking about better than they can.

The conference will end on Friday with Strominger and Ooguri on *The Future of String Theory*. As at all string theory conferences, they surely will explain how string theorists deserve an A+++, great progress is being made, the future is bright, etc. They have put together a list of 100 open questions. Number 83 asks what will happen now that the founders of string theory are retiring and dying off, suggesting that AI is the answer:

train an LLM with the very best papers written by the founding members, so that it can continue to set the trend of the community.

That’s all I can stand of this kind of thing for now without getting hopelessly depressed about the future. I’ll try in coming weeks to write more about very different topics, and stop wasting time on the sad state of affairs of a field that long ago entered intellectual collapse.

]]>I had thought that the universally negative reaction to the fall 2022 wormhole publicity stunt meant that we’d never hear more about this, with even the editors of Quanta magazine having understood that they’d been had. While away on vacation though, I learned from Dulwich Quantum Computing that all the authors of the original stunt are back, now claiming not just wormhole teleportation, but Long-range wormhole teleportation.

I’d also thought that no one at this point could possibly think it was a good idea to help these authors go to the public with their claims about creating wormholes in a lab. It seems though that this coming weekend if you’re here in NYC you can buy tickets to listen to some of them explain in person

]]>the mind-bending speculation that we may be able to create wormholes—tunnels through spacetime—in the laboratory.

The semester here is coming to a close. I’m way behind writing up notes for the lectures I’ve been giving, which are ending with covering the details of the Standard Model. This summer I’ll try to finish the notes and will be working on writing out explicitly the details of how the Standard Model works in the “right-handed” picture of the spinor geometry of spacetime that I outlined here.

At this point I need a vacation, heading soon to France for a couple weeks, then will return here and get back to work. There may be little to no blogging here for a while.

On the Langland’s front, Laurent Fargues is turning his Eilenberg lectures here last fall into a book, available here. In Bonn, Peter Scholze is running a seminar on Real local Langlands as geometric Langlands on the twistor-P1

]]>Until about a year and a half ago, the way to get funding in physics was to somehow associate yourself to the hot trend of quantum computing and quantum information theory. Large parts of the string theory and quantum gravity communities did what they could to take advantage of this. On November 30, 2022, this all of a sudden changed as two things happened on the same day:

- Quanta magazine, Nature and various other places were taken in by a publicity stunt, putting out that day videos and articles about how “Physicists Create a Wormhole Using a Quantum Computer”. The IAS director compared the event to “Eddington’s 1919 eclipse observations providing the evidence for general relativity.” Within a few days though, people looking at the actual calculation realized that these claims were absurd. The subject had jumped the shark and started becoming a joke among serious theorists. That quantum computers more generally were not living up to their hype didn’t help.
- OpenAI released ChatGPT, very quickly overwhelming everyone with evidence of how advanced machine learning-based AI had become.

If you’re a theorist interested in getting funding, obviously the thing to do was to pivot quickly from quantum computing to machine learning and AI, and get to work on the people at Quanta to provide suitable PR. Today Quanta features an article explaining how “Using machine learning, string theorists are finally showing how microscopic configurations of extra dimensions translate into sets of elementary particles.”

Looking at these new neural network calculations, what’s remarkable is that they’re essentially a return to a failed project of nearly 40 years ago. In 1985 the exciting new idea was that maybe compactifying a 10d superstring on a Calabi-Yau would give the Standard Model. It quickly became clear that this wasn’t going to work. A minor problem was that there were quite a few classes of Calabi-Yaus, but the really big problem was that the Calabi-Yaus in each class were parametrized by a large dimensional moduli space. One needed some method of “moduli stabilization” that would pick out specific moduli parameters. Without that, the moduli parameters became massless fields, introducing a huge host of unobserved new long-range interactions. The state of the art 20 years later is that endless arguments rage over whether Rube Goldberg-like constructions such as KKLT can consistently stabilize moduli (if they do, you get the “landscape” and can’t calculate anything anyway, since these constructions give exponentially large numbers of possibilities).

If you pay attention to these arguments, you soon realize that the underlying problem is that no one knows what the non-perturbative theory governing moduli stabilization might be. This is the “What Is String Theory?” problem that a consensus of theorists agrees is neither solved nor on its way to solution.

The new neural network twist on the old story is to be able to possibly compute some details of explicit Calabi-Yau metrics, allowing you to compute some numbers that it was clear back in the late 1980s weren’t really relevant to anything since they were meaningless unless you had solved the moduli stabilization program. Quanta advertises this new paper and this one (which “opens the door to precision string phenomenology”) as well a different sort of calculation which used genetic algorithms to show that “the size of the string landscape is no longer a major impediment in the way of constructing realistic string models of Particle Physics.”

I’ll end with a quote from the article, in which Nima Arkani-Hamed calls this work “garbage” in the nicest possible way:

“String theory is spectacular. Many string theorists are wonderful. But the track record for qualitatively correct statements about the universe is really garbage,” said Nima Arkani-Hamed, a theoretical physicist at the Institute for Advanced Study in Princeton, New Jersey.

A question for Quanta: why are you covering “garbage”?

]]>A few items on the science outreach front:

- The Oscars of Science were held Saturday night in Hollywood, with a long list of A-listers in attendance, led by Kim Kardashian. More here, here and here.
You’ll be able to watch the whole thing on Youtube starting April 21.

- The World Science Festival will have some live programs here in New York May 30 – June 2. One of the programs will feature the physicists responsible for the Wormhole Publicity Stunt explaining how

we may be able to create wormholes—tunnels through spacetime—in the laboratory.

- Stringking42069 is back on Twitter with his outreach efforts for the string theory community.

This semester the KITP has been running a program asking What is String Theory?, which is winding up next week, and was promising to “arrive at a deeper answer to the question in the title.” It seems though that this effort has gone nowhere, with this report from the scene:

Went to a string theory conference with many of the top researchers in the field centered around tackling the question “what is string theory” and the consensus after the conference was that nobody knows lmao

For an answer to the question from someone with a lot more experience, I recently noticed that Lubos Motl is very active on Quora, giving thousands of sensible answers to a range of questions, especially having to do with Central Europe. He explains the relation of string theory and M-theory (disagreeing with Wikipedia), and defines string theory as

the name of the consistent theory of quantum gravity which covers all the vacua found in the context of critical string theory and M-theory.

I had trouble getting my head around the concept of an undefined theory known to be consistent when I first heard about it nearly 30 years ago, but it seems to still be a thing.

]]>Sabine Hossenfelder today posted a new video on youtube which everyone in theoretical physics should watch and think seriously about. She tells honestly in detail the story of her career and experiences in academia, explaining very clearly exactly what the problems are with the conventional system for funding research and for training postdocs.

After a string of postdocs requiring moving and living far from her husband, she decided she needed to move back to Germany and applied for a grant to fund her research (I believe for this project). This is how she describes the situation:

At this point I’d figured out what you need to put into a grant proposal to get the money. And that’s what I did. I applied for grants on research projects because it was a way to make money, not because I thought it would leave an impact in the history of science. It’s not that was I did was somehow wrong. It was, and still is, totally state of the art. I did what I said I’d do in the proposal, I did the calculation, I wrote the paper, I wrote my reports, and the reports were approved. Normal academic procedure.

But I knew it was bullshit just as most of the work in that area is currently bullshit and just as most of academic research that your taxes pay for is almost certainly bullshit. The real problem I had, I think, is that I was bad at lying to myself. Of course, I’d try to tell myself and anyone who was willing to listen that at least unofficially on the side I would do the research that I thought was worth my time but that I couldn’t get money for because it was too far off the mainstream. But that research never got done because I had to do the other stuff that I actually got paid for.

As that grant ended, she decided to try instead applying for grants to work on research that she found to be more promising and not bullshit, but those grant proposals were not successful. Since then, she has left the academic research system and concentrated on trying to make a career oriented around high-quality Youtube videos about scientific research.

It seems to me that Hossenfelder correctly analyzes the source of her difficulties: “The real problem I had, I think, is that I was bad at lying to myself.” Those more successful in the academic system sometimes criticize her as someone just not as talented as themselves at recognizing and doing good research work. But I see quite the opposite in her story. Many of those successfully pursuing a research career in this area differ from her in either not being smart enough to recognize bullshit, or not being honest enough to do anything about it when they do recognize bullshit.

]]>I don’t really have time to write seriously about this, and there’s a very good argument that this is a topic anyone with any sense should be ignoring, but I just can’t resist linking to the latest in the abc saga, the REPORT ON THE RECENT SERIES OF PREPRINTS BY K. JOSHI posted yesterday by Mochizuki.

To summarize the situation before yesterday, virtually all experts in this subject have long ago given up on the idea that Mochizuki’s IUT theory has any hope of proving the abc conjecture. Back in 2018, after a trip to Kyoto to discuss in depth with Mochizuki, Scholze and Stix wrote up a document explaining why the IUT proof strategy was flawed. Scholze later defended this argument in detailand as far as I know has not changed his mind. Taking a look at these two documents and at Mochizuki’s continually updated attempt to refute them, anyone who wants to try and decide for themselves can make up their own minds. All experts I’ve talked to agree that Scholze/Stix are making a credible argument, Mochizuki’s seriously lacks credibility.

The one hope for an IUT-based proof of abc has been the ongoing work of Kirti Joshi, who recently posted the last in a series of preprints purporting to give a proof of abc, starting off with “This paper completes (in Theorem 7.1.1) the remarkable proof of the abc-conjecture announced by Shinichi Mochizuki…”. My understanding is that Scholze and other experts are so far unconvinced by the new Joshi proof, although I don’t know of anyone who has gone through it carefully in detail. Given this situation, an IUT optimist might hope that the Joshi proof might work and vindicate IUT.

Mochizuki’s new report destroys any such hope, simultaneously taking a blow-torch to his own credibility. He starts off with

.. it is

conspicuously obviousto any reader of these preprints who is equipped with a solid, rigorous understanding of the actual mathematical content of inter-universal Teichmüller theory that the author of this series of preprints isprofoundly ignorantof the actual mathematical content of inter-universal Teichmüller theory, and, in particular, that this series of preprintsdoes not contain, at least from the point of view of the mathematics surrounding inter-universal Teichmüller theory,any meaningful mathematical content whatsoever.

and it gets worse from there.

]]>David Tong has produced a series of very high quality lectures on theoretical physics over the years, available at his website here. Recently a new set of lectures has appeared, on the topic of the Standard Model. Skimming through these, they look quite good, with explanations that are significantly more clear than found elsewhere.

Besides recommending these for their clarity, I can’t help pointing out that there is one place early on where the discussion is confusing, at exactly the same point as in most textbooks, and exactly at the point that I’ve been arguing that something interesting is going on. On page 7 of the notes we’re told

We can, however, ﬁnd two mutually commuting $\mathfrak{su}(2)$ algebras sitting inside $\mathfrak{so}(1, 3)$.

but this is true only if you complexify these real Lie algebras. What’s really true is

$$\mathfrak{so}(1, 3)\otimes \mathbf C = (\mathfrak{su}(2)\otimes \mathbf C) + (\mathfrak{su}(2)\otimes \mathbf C)$$

Note that

$$\mathfrak{su}(2)\otimes \mathbf C=\mathfrak{sl}(2,\mathbf C)$$

Tong is aware of this, writing on page 8:

The Lie algebra $\mathfrak{so}(1, 3)$ does not contain two, mutually commuting copies of the real Lie algebra $\mathfrak{su}(2)$, but only after a suitable complexiﬁcation. This means that certain complex linear combinations of the Lie algebra $su(2)\times su(2)$ are isomorphic to $so(1, 3)$. To highlight this, the relationship between the two is sometimes written as

$$\mathfrak{so}(1, 3) \equiv \mathfrak{su}(2) \times \mathfrak{su}(2)^*$$

This is a rather confusing formula. What it is trying to say is that the real Lie algebra $\mathfrak{so}(3,1)$ is the conjugation invariant subspace of its complexification

$$(\mathfrak{su}(2)\otimes \mathbf C) + (\mathfrak{su}(2)\otimes \mathbf C)$$

where the conjugation interchanges the two factors. Tong goes on to use this to identify conjugating an $\mathfrak{so}(3,1)$ representation with interchanging its properties as representations of the two $\mathfrak{su}(2)\otimes \mathbf C=\mathfrak{sl}(2,\mathbf C)$ factors.

For a very detailed explanation of the general story here, involving not just the Lorentz real form of the complexification of $\mathfrak{so}(3,1)$, but also the other (Euclidean and split signature) real forms, see chapter 10 of the notes here. My “spacetime is right-handed” proposal is that instead of identifying the physical Lorentz Lie algebra in the above manner as the “anti-diagonal” sub-algebra of the complexification, one should identify it instead with one of the two $\mathfrak{sl}(2,\mathbf C)$ factors (calling it the “right-handed” one).

]]>I was very pleased to hear yesterday that this year’s Abel Prize has been awarded to Michel Talagrand. For more about Talagrand and his mathematics, see the Abel site, Quanta, NYT, Nature and elsewhere. Also, see lots of reactions on Twitter like this one.

Almost exactly ten years ago I got an email from someone whose name I didn’t recognize, expressing interest in the notes I had made available online which would turn into the book on quantum mechanics. He was reading the notes and had some comments which he included, saying he thought they were trivial but maybe I would want to take a look. Some of them were of the type “I don’t quite understand the argument on page X”. Figuring that I’d help out an earnest reader with a weak background by explaining the argument a bit better, I took a look at the argument on page X. After a while I realized that what I had written was nonsense, a very different argument was needed. “I don’t quite understand” was his way of politely telling me “you have this completely wrong.”

I soon ran into Yannis Karatzas and asked him if he knew anything about this “Michel Talagrand”. He told me “of course! He’s amazing, almost got a Fields Medal”. Over the next year or two I benefited tremendously from Michel continuing to read carefully through my notes and send me detailed comments. He was very much responsible for improving a lot the quality and accuracy of what I was writing. He had begun his own project of trying to understand quantum field theory by writing a book about it. The result is available as What Is a Quantum Field Theory?, which is a wonderful resource for anyone interested in a precise and accurate account of much of the basics of the subject. If you’ve seen Gerald Folland’s excellent Quantum Field Theory: A Tourist Guide for Mathematicians, you can think of Talagrand’s book as a much expanded version, giving the full story that Folland only sketched.

During many of my trips to Paris since that time I’ve gotten together with Michel and his wife Wansoo, and have also seen them here in New York. It has been a great pleasure to get to know Michel in person, he is a wonderful human being as well as a truly great mathematician.

]]>The first entry on this blog was 20 years ago yesterday, first substantive one was 20 years ago tomorrow (first one that drew attacks on me as an incompetent was two days later). Back when I started this up, blogging was all the rage, and lots of other blogs about fundamental physics were starting around the same time. Almost all of these have gone dormant, with Sabine Hossenfelder’s Backreaction one notable exception. She and some others (like Sean Carroll) have largely moved to video, which seems to be the thing to do to communicate with as many people as possible. There are people who do “micro-blogging” on Twitter, with the descendant of Lubos Motl’s blog StringKing42069 on Twitter. I remain mystified why anyone thinks it’s a good idea to discuss complex issues of theoretical physics in the Twitter format, flooded with all sorts of random stupidity.

Looking back on what I was writing 20 years ago it seems to me to have held up well, and there is very little that I would change. The LHC experiments have told us that the Standard Model Higgs is there, and that supersymmetry is not, but these were always seen as the most likely results.

My point of view on things has changed since then, especially in recent years. When I started the blog I was 20 years past my Ph.D., in the middle of some sort of an odd career. Today I’m 66, 40 years past the Ph.D., much closer to the end of a career and a life than to a beginning. In 2004 I was looking at nearly twenty years of domination of fundamental theory by a speculative idea that to me had never looked promising and by then was clearly a failure. 20 years later this story has become highly disturbing. The refusal to admit failure and move on has to a large degree killed off the field as a serious science.

The technical difficulties involved in reaching higher energy scales at this point makes it all too likely that I’m not going to see any significant new data about what the world looks like above the TeV scale during my lifetime. Without experiment to keep it honest, fundamental theory has seriously gone off the rails in a way which looks to me irreparable. With the Standard Model so extremely successful and no hints from experiment about how to improve it, it’s now been about 50 years that this has been a subject in which it is very difficult to make progress. I’ve always been an admitted elitist: in the face of a really hard problem, only a very talented person trained as well as possible and surrounded by the right intellectual environment is likely to be able to get somewhere.

My background has been at the elite institutions that are supposed to be providing this kind of training and working environment. Harvard and Princeton provided my training in 1975-1984 and I think did a good job of it at the time, but things are now quite different. 40 years of training generations of students in a failed research program has taken its toll on the subject. I remember well what it was like to be an ambitious student at these places, determined to get as quickly as possible to the frontiers of knowledge, which in those times meant learning gauge field theory. These days it unfortunately means putting a lot of effort into reading Polchinski, and becoming expert in the technology of failed ideas.

One recent incident that destroyed my remaining hopes for the institutions I had always still had some faith in was the program discussed here, which made me physically ill. It made it completely clear that the leaders of this subject will never admit what has happened, no matter how bad it gets. Also having a lot of impact on me was the Wormhole Publicity Stunt, which showed that the problem is not just refusing to face up to the past, but willingness to sign onto an awful view of the future, as long as it brings in funding and can be sold as vindication of the past. Watching the director of the IAS explain that this was comparable to the 1919 vindication of GR surely made more than a few of those in attendance at least queasy. This particular stunt may have jumped the shark, but what’s likely coming next looks no better (replace quantum computing with AI).

The strange thing is that while the wider world and the subject I care most about have been descending into an ever more depressing environment of tribalistic behavior and intellectual collapse, on a personal level things are going very well. In particular I’m ever more optimistic about some new ideas and enjoying trying to make progress with them, seeing several promising directions. Whatever years I have available to think about these things are looking like they should be intellectually rewarding ones. Locally, I’m looking forward to what the next twenty years will bring (if I make it through them…), while on a larger scale I’m dreading seeing what will happen.

]]>This spring I’ve been teaching a course aimed at math graduate students, starting with quantum mechanics and trying to get to an explanation of the Standard Model by the end of the semester. Course notes for the first half of the course are available here, but still quite preliminary, in particular I need to do a lot of work on section 9.4 and add material to chapter 10. Hoping to get to this tomorrow.

There won’t be much progress on the notes for the next couple weeks. This coming week I’m hoping to spend some time trying to understand Peter Scholze’s IAS lectures, will go down to the IAS on Tuesday. On Thursday I’m heading out on a spring break vacation to the Arizona-Utah desert.

Perhaps a good way to think about these notes is that they’re both aimed more at mathematicians than physicists (although I hope accessible to many physicists) and also designed more to supplement than to replace the discussions in the standard physics texts. So, a lot of the standard material is not there, since it’s well-covered elsewhere, but there are a lot of topics covered that usually aren’t.

One unusual aspect of the notes is that I spend a lot of time trying to explain non-relativistic quantum field theory, since that seems to me to be a better starting point than immediately diving into the relativistic case. I’d be curious to know if anyone can point me to a good discussion of the path integral formalism for non-relativistic quantum field theory, which is something I haven’t found. This is one reason it’s taking a while to finish writing up my own version.

Also original here I think is a careful discussion of the real forms of spinors and twistors. This in some sense is background for the new ideas about “spacetime is right-handed” which I’ve been working on. Nothing in the notes now about the new ideas, but I hope the explanation of the conventional story in these notes is useful.

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

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

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

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

- Dan Garisto has a very good article here examining the present state of high energy experimental particle physics and phenomenology. He also summarizes current thoughts about the future. The CERN FCC-ee proposal is still in feasibility study mode, with the big problem its high cost. Numbers like \$15 – \$20 billion have shown up in press reports, and Garisto has “tens of billions”. The feasibility study is supposed to be finished late next year, and presumably a big part of it is people at CERN crunching numbers trying to figure out some plausible way this could work financially.
- There’s a very good article at Quanta about efforts to put together version 3 of the very influential “Kirby List” of open problems in topology. For version 1, see here, version 2 here, and as far as I can tell, version 3 still not finished (but discussed here).
- Brian Keating has a video asking “What would It take for String Theory to move beyond the realm of pure math, Into verifiable territory”? The answer obviously is a conventional substantive scientific prediction of anything. By describing the problem with string theory as being that it is now in “the realm of pure math”, I think Keating is missing the point. The problem with “string theory” is not that it’s pure math, but that it’s not a theory. “String theory” is now a 50 year old set of failed hopes and dreams that a theory might exist, see for instance Theorists Without a Theory. Every so often I try to figure out what people still pursuing this are up to, most recently today taking a quick look at KITP talks by Liam McAllister here and here.
The problem shows up clearly at 10:23 of the first talk where he’s talking about his goal, which gives up on studying all or even representative string theory solutions and tries just to find any “valid solutions”. But what is a “valid solution”?

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

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

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

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

- This spring I’ve been teaching a graduate course aimed at getting to an explanation of the Standard Model aimed at mathematicians. The first few weeks have been about quantization and quantum mechanics, today I’m starting on quantum field theory, starting with developing the framework of non-relativistic QFT. While trying to figure out how best to pass from QM to QFT, I’ve kept coming across various aspects of this that I’ve always found confusing, never seen a good explanation of. Today I ran across a wonderful article by Thanu Padmanabhan, who I knew about just because of his very good introductory book on QFT, Quantum Field Theory: The why, what and how. The article is called “Obtaining the Non-relativistic Quantum Mechanics from Quantum Field Theory: Issues, Folklores and Facts” and subtitled “What happens to the anti-particles when you take the non-relativistic limit of QFT?” It contains a lot of very clear discussion of issues that come up when you try and think about the QM/QFT relationship, a sort of thing I haven’t seen anywhere else.
Looking for more of Padmanabhan’s writings, I was sad to find out that he passed away in 2021 at a relatively young age, which is a great loss. For more about him, there’s a collection of essays by those who knew him available here.

- For something I can’t recommend paying more attention to, New Scientist has an article labeled How to test String Theory, which is mostly an interview with Joseph Conlon. Conlon’s goal is to make the case for string theory, in its original form as a unified theory with compactified extra dimensions. On the issue of testability there’s nothing new, just the usual unfalsifiable story that among all of the extra stuff (moduli fields, axions, extra dimension, extended structures) that appears in string theory and that string theorists have to go to great trouble to make non-observable, it’s in principle conceivable that somebody might observe one of these things someday. But, that’s not really what people mean when they ask for a test. For details of the sort of thing he’s talking about in the New Scientist article, see here.
I strongly disagree with Conlon about some of what he’s saying, but the situation is very much like it has always been with many string theorists since way back to nearly forty years ago. We don’t disagree about the facts, it’s just that I’ve always looked at these facts and interpreted them as showing string theory unification ideas to be unpromising, whereas string theorists like Conlon somehow find reasons for optimism, or at least for believing there’s no better thing to do with their time. Last time I was in Oxford, Conlon invited me to lunch at his College and I enjoyed our conversation. I think we agreed on many topics, even about what is going on in string theory, but it looks like we’re always going to have diametrically opposed views on this particular question. For more from him, as far as popular books by string theorists go, his Why String Theory? book is about the best there is, see more about this here.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

The article ends with:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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