The Mystery of Spin

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
$$M(2,\mathbf C)=\mathbf H\otimes_{\mathbf R}\mathbf C$$
Note added: complexified quaternions are often called “biquaternions”
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 $M(2,\mathbf C)$.

Update: There’s a very new video here, where Keith Conrad discusses quaternions, especially the case of quaternion algebras over $\mathbf Q$ and their relation to quadratic reciprocity.

Posted in Uncategorized | 15 Comments

Strings 2024

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.

Update: The slides for the AI talk are here. The speaker is Jared Kaplan, a Johns Hopkins theorist who is a co-founder of Anthropic and on leave working as its Chief Science Officer. His talk has a lot of generalities about AI and its very fast progress, little specifically about AI doing theoretical physics.

Posted in Strings 2XXX | 31 Comments

Wormholes, Part Deux

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.

Posted in Wormhole Publicity Stunts | 4 Comments

Various and Sundry

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

Update: One more item. Videos of talks from a conference on arithmetic geometry in honor of Helene Esnault at the IHES last week are now available. Dustin Clausen’s talk covers one of my favorite topics (the Cartan model for equivariant cohomology), making use of the new formalism for handling he has developed with Scholze for handling C-infinity manifolds in a more algebraic way.

Update: Now back from vacation. While I was away, Quanta made up for its nonsense like this with a very nice article about “Weil’s Rosetta Stone” and what it has to do with geometric Langlands. In the comments people have pointed to the proof of geometric Langlands that has finally been finished, and New Scientist has an article (or see Edward Frenkel on Twitter here).

Posted in Euclidean Twistor Unification, Langlands | 14 Comments

This Week’s Hype

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”?

Update: String theorist Marcos Mariño on twitter:

In my view, using today’s AI to calculate the details of string compactifications is such a waste of time that I fear that a future Terminator will come to our present to take revenge for the merciless, useless exploitation of its grandparents.

Update: More string theory AI hype here.

Posted in This Week's Hype | 23 Comments

Science Outreach News

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.
Posted in Uncategorized, Wormhole Publicity Stunts | 27 Comments

What is String Theory?

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.

Posted in Uncategorized | 25 Comments

How I fell out of love with academia

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 what 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.

Posted in Uncategorized | 35 Comments

A Report From Mochizuki

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 detail and 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 obvious to 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 is profoundly ignorant of the actual mathematical content of inter-universal Teichmüller theory, and, in particular, that this series of preprints does 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.


Update
: A commenter points to a response from Joshi here.

Update: Scholze has a comment on MathOverflow indicating precisely where Joshi’s attempted proof runs into trouble.

Update: Mochizuki and those around him award themselves \$100,000 (this is the IUT Innovator Prize described here).

Posted in abc Conjecture | 51 Comments

David Tong: Lectures on the Standard Model

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). Conjugation on representations is then just the usual conjugation of representations of the right-handed $\mathfrak{sl}(2,\mathbf C)$ factor.

Posted in Euclidean Twistor Unification, Uncategorized | 13 Comments