Wick Rotating Weyl Spinor Fields

Posted on December 7, 2024 by woit

It’s been taking me forever to sort out and write down the details of implications of the proposal described here. While waiting for that to be done, I thought it might be a good idea to write up one piece of this, which might be some sort of introductory part of the long document I’ve been working on. This at least starts out very simply, explaining what is going on in terms that should be understandable by anyone who has studied the quantization of a spinor field.

I’m not saying anything here about how to use this to get a better unified theory, but am pointing to the precise place in the standard QFT story (the Wick rotation of a Weyl degree of freedom) where I see an opportunity to do something different. This is a rather technical business, which I’d love to convince people is worth paying attention to. Comments from anyone who has thought about this before extremely welcome.

Matter degrees of freedom in the Standard Model are described by chiral spinor fields. Before coupling to gauge fields and the Higgs, these all satisfy the Weyl equation

$$(\frac{\partial}{\partial t}+\boldsymbol\sigma\cdot\boldsymbol\nabla)\psi (t,\mathbf x)=0$$

The Fourier transform of this equation is

$$ (E-\boldsymbol \sigma\cdot \mathbf p)\widetilde{\psi}(E,\mathbf p)=0$$

Multiplying by $(E+\boldsymbol \sigma\cdot \mathbf p)$, solutions satisfy

$$(E^2-|\mathbf p|^2)=0$$

so are supported on the positive and negative light-cones $E=\pm |\mathbf p|$.



The helicity operator

$$\frac{1}{2}\frac{\boldsymbol\sigma\cdot \mathbf p}{|\mathbf p|}$$

will act by $+\frac{1}{2}$ on positive energy solutions, which are said to have “right-handed” helicity. For negative energy solutions, the eigenvalue will be $-\frac{1}{2}$ and these are said to have “left-handed helicity”.



The quantized field $\widehat{\psi}$ will annihilate right-handed particles and create left-handed anti-particles, while its adjoint $\widehat{\psi}^\dagger$ will create right-handed particles and annihilate left-handed anti-particles. One can describe all the Standard model matter particles using such a field. Particles like the electron which have both right-handed and left-handed components can be described by two such chiral fields (note that one is free to interchange what one calls a “particle” or “anti-particle”, or equivalently, which field is $\widehat{\psi}$ and which is the adjoint). Couplings to gauge fields are introduced by changing derivatives to covariant derivatives.



The Lagrangian will be

\begin{equation}
\label{eq:minkowski-lagrangian}
L=\psi^\dagger(\frac{\partial}{\partial t}+\boldsymbol\sigma\cdot\boldsymbol\nabla)\psi

\end{equation}

which is invariant under an action of the group $SL(2,\mathbf C)$, the spin double-cover of the time-orientation preserving Lorentz transformations. To see how this works, note that one can identify Minkowski space-time vectors with two dimensional self-adjoint complex matrices, as in

$$(E,\mathbf p)\leftrightarrow M(E,\mathbf p)=E-\boldsymbol \sigma\cdot \mathbf p=\begin{pmatrix} E-p_3& -p_1+ip_2\\-p_1-ip_2&E+p_3\end{pmatrix}$$

with the Minkowski norm-squared $-E^2-|\mathbf p|^2=-\det M$. 
Elements $S\in SL(2,\mathbf C)$ act by

$$M\rightarrow SMS^\dagger$$

which, since it preserves self-adjointness and the determinant, is a Lorentz transformation.



The propagator of a free chiral spinor field in Minkowski space-time is (like other qfts) ill-defined as a function. It is a distribution, generally defined as a certain limit ($i\epsilon$ prescription). This can be done by taking the time and energy variables to be complex, with the propagator a function holomorphic in these variables in certain regions, giving the real time distribution as a boundary value of the holomorphic function. One can instead “Wick rotate” to imaginary time, where the analytically continued propagator becomes a well-defined function.



There is a well-developed formalism for working with Wick-rotated scalar fields in imaginary time, but Wick-rotation of a chiral spinor field is highly problematic. The source of the problem is that in Euclidean signature spacetime, the identification of vectors with complex matrices works differently. Taking the energy to be complex (so of the form $E+is$), Wick rotation gives matrices

$$\begin{pmatrix} is-p_3& -p_1+ip_2\\-p_1-ip_2&is+p_3\end{pmatrix}$$

which are no longer self-adjoint. The determinant of such a matrix is minus the Euclidean norm-squared $(s^2 +|\mathbf p|^2)$. Identifying $\mathbf R^4$ with matrices in this way, the spin double cover of the orthogonal group $SO(4)$ is

$$Spin(4)=SU(2)_L\times SU(2)_R$$

with elements pairs $S_L,S_R$ of $SU(2)$ group elements, acting by

$$\begin{pmatrix} is-p_3& -p_1+ip_2\\-p_1-ip_2&is+p_3\end{pmatrix}\rightarrow S_L\begin{pmatrix} is-p_3& -p_1+ip_2\\-p_1-ip_2&is+p_3\end{pmatrix}S_R^{-1}$$

The Wick rotation of the Minkowski spacetime Lagrangian above will only be invariant under the subgroup $SU(2)\subset SL(2,C)$ of matrices such that $S^\dagger=S^{-1}$ (these are the Lorentz transformations that leave the time direction invariant, so are just spatial rotations). It will also not be invariant under the full $Spin(4)$ group, but only under the diagonal $SU(2)$ subgroup. The conventional interpretation is that a Wick-rotated spinor field theory must contain two different chiral spinor fields, one transforming undert $SU(2)_L$, the other under $SU(2)_R$.

The argument of this preprint is that it’s possible there’s nothing wrong with the naive Wick rotation of the chiral spinor Lagrangian. This makes perfectly good sense, but only the diagonal $SU(2)$ subgroup of $Spin(4)$ acts non-trivially on Wick-rotated spacetime. The rest of the $Spin(4)$ group acts trivially on Wick-rotated spacetime and behaves like an internal symmetry, opening up new possibilities for the unification of internal and spacetime symmetries.

From this point of view, the relation between spacetime vectors and spinors is not the usual one, in a way that doesn’t matter in Minkowski spacetime, but does in Euclidean spacetime. More specifically, in complex spacetime the Spin group is

$$Spin(4,\mathbf C)=SL(2,\mathbf C)_L\times SL(2,\mathbf C)_R$$

there are two kinds of spinors ($S_L$ and $S_R$) and the usual story is that vectors are the tensor product $S_L\otimes S_R$. Restricting to Euclidean spacetime all that happens is that the $SL(2,\mathbf C)$ groups restrict to $SU(2)$. 



Something much more subtle though is going on when one restricts to Minkowski spacetime. There the usual story is that vectors are the subspace of $S_L\otimes S_R$ invariant under the action of simultaneously swapping factors and conjugating. These are acted on by the restriction of $Spin(4,\mathbf C)$ to the $SL(2,\mathbf C)$ anti-diagonal subgroup of pairs $(\Omega,\overline{\Omega})$.

The proposal here is that one should instead take complex spacetime vectors to be the tensor product $S_R\otimes \overline{S_R}$, only using right-handed spinors, and the restriction to the Lorentz subgroup to be just the restriction to the $SL(2,\mathbf C)_R$ factor. This is indistinguishable from the usual story if you just think about Minkowski spacetime, since then all you have is one $SL(2,\mathbf C)$, its spin representation $S$ and the conjugate $\overline S$ of this representation.
Exactly because of this indistinguishability, one is not changing the symmetries of Minkowski spacetime in any way, in particular not introducing a distinguished time direction.



When one goes to Euclidean spacetime however, things are quite different than the usual story. Now only the $SU(2)_R$ subgroup of $Spin(4)=SU(2)_L\times SU(2)_R$ acts non-trivially on vectors, the $SU(2)_L$ becomes an internal symmtry. Since $S_R$ and $\overline{S_R}$ are equivalent representations, the vector representation is equivalent to $S_R\otimes S_R$ which decomposes into the direct sum of a one-dimensional representation and a three-dimensional representation. Unlike in Minkowski spacetime there is a distinguished direction, the direction of imaginary time.



Having such a distinguished direction is usually considered to be fatal inconsistency. It would be in Minkowski spacetime, but the way quantization in Euclidean quantum field theory works, it’s not an inconsistency. To recover the physical real time, Lorentz invariant theory, one need to pick a distinguished direction and use it (“Osterwalder-Schrader reflection”) to construct the physical state space.

 Besides the preprint here, see chapter 10 of these notes for a more detailed explanation of the usual story of the different real forms of complexified four-dimensional space.


Posted in Euclidean Twistor Unification | 5 Comments

This Week’s Hype

Posted on November 22, 2024 by woit

If a post-truth field of science is going to keep going, it needs to convince funders and the public that progress is being made, so there’s a continual need for people uninterested in truth and willing to produce appropriate propaganda. This is the 142nd edition of This Week’s Hype, which has been documenting this phenomenon for the past twenty years.

Such a post-truth project requires cooperation from institutions responsible for communicating science to the public. One such is the Royal Institution which sponsored a program of pure propaganda for string theory, now available on Youtube. From the transcript:

I’m not in propaganda mode here, and we shall avoid propaganda mode… As you see, I’m trying not to go into propaganda mode… Once again I’m in no propaganda mode, but we are fairly sure…

If a speaker four times in a talk assures you that what he’s saying is not propaganda, one thing you can be sure of is that it is propaganda.

Another part of maintaining a post-truth scientific field is that you need people willing to write propaganda “scientific” articles, institutions willing to publish such articles and venues to promote them. A good example of this is The Standard Model from String Theory: What Have We Learned? now published in The Annual Review of Nuclear and Particle Science.

The publisher of Annual Reviews has a publication called Knowable Magazine, tasked with promoting their articles, and they’ve hired Tom Siegfried to write about this one under the title String theory is not dead. By the way, if somebody is hiring journalists to write propaganda pieces entitled “Field X is not dead”, you can be sure that field X truly is dead. Siegfried has had a very long career in the string theory propaganda business, going back nearly 30 years. See for instance this posting, which has some background on Siegfried.

In his very hostile review of Not Even Wrong for the New York Time, Siegfried explains that I’m completely wrong about string theory’s lack of predictions:

…string theory does make predictions — the existence of new supersymmetry particles, for instance, and extra dimensions of space beyond the familiar three of ordinary experience. These predictions are testable: evidence for both could be produced at the Large Hadron Collider, which is scheduled to begin operating next year near Geneva.

Like all of those in the post-truth business, having one’s “predictions” turn out to not work doesn’t have any impact at all on one’s willingness to keep the propaganda campaign going.

A good giveaway that something is propaganda is a title that indicates that you’re not going to get just information about something, but also a sales job. Today the Higgs Centre in Edinburgh has a talk scheduled with the title What is string theory and why you should care?. The idea that people at a theoretical physics center would not know what string theory is after the past forty years is pretty laughable, so clearly the point of this talk is not the first part of the title, but the “you should care” part.

Update: Video of the Higgs Centre talk by string theorist Sašo Grozdanov is now available here. As usual in such things, lots of discussion of the quantization of the single-quantized theory of a bosonic string, which connects not at all to physics. No discussion of why the much more complicated things you would need to do to try and make this look like physics simply don’t work. Grozdanov’s acknowledges criticism of string theory, but claims that it’s just “sociological”, coming from people who are too impatient. According to him (and he says he’s embodying the consensus of the field):

  • “It’s the only way forward”
  • “We have nothing else”
  • “It’s the only thing that works”

He acknowledges there’s no connection to the real world, interprets this though as only indicating that “we’re missing something” (since alternatives are not conceivable).

Posted in This Week's Hype | 6 Comments

The Impossible Man

Posted on November 15, 2024 by woit

There’s a new book out this week, a biography of Roger Penrose by Patchen Barss, with the title The Impossible Man: Roger Penrose and the cost of genius. Penrose is one of the greatest figures in physics and mathematical physics of the second half of the twentieth century, arguably the dominant theorist in the field of general relativity. His work on twistors is the most important new idea about space-time geometry post-Einstein, and I believe it will be studied long after string theory has been finally consigned to the oblivion of failed ideas. His 2004 book The Road to Reality is an unparalleled comprehensive summary of the geometric point of view on fundamental physics, a huge work of genius written to try and convey the deepest ideas around to as many people as possible.

The new biography provides a lot of detail about Penrose’s life and work, well beyond what I’d learned over the years from reading his writings and those of others who worked with him. It does a good job of explaining to a wide audience some areas of his work, and how the background he grew up in helped make some of his great achievements possible. From an early age, Penrose was fascinated by geometry, and he became our greatest master at visualizing four dimensional space-time, generating deep insights into the subject. While one can motivate twistor theory in several very different ways, it came to him through such visualization.

Another thing I learned from the book was more of the story of the singularity theorems for which he was awarded the Nobel Prize in 2020. While Hawking often gets more attention for this, it seems that there’s a good case that the creative ideas there were more Penrose’s, with Hawking much better at getting attention for his work. That, despite having read a great deal about this story over the years, I’d never heard that Penrose saw things this way until reading this book is much to his credit.

In later parts of the book, the author handles well the issue of some of Penrose’s more problematic later projects. Experts on cosmology are highly skeptical of his conformal cyclic cosmology ideas, and pretty much everyone thinks his involvement with Stuart Hameroff around questions having to do with consciousness has been misguided.

Penrose played an important role in my life, by suggesting to his publisher that they publish Not Even Wrong (for the story of that, see here). While, I haven’t been in contact with him for many years, and only have met him in person briefly twice, he seemed to me unassuming and more likely to be friendly and helpful to others than your average academic.

Unfortunately, the book pairs a largely very good discussion of Penrose’s scientific career with a very extensive and rather unsympathetic discussion of his personal life. If you read reviews such as the one today in the Wall Street Journal, you’ll be told that Penrose’s personal story “fits the template” of the genius as “deeply weird”, with the book showing that “the cost of genius” is personal sacrifices by those around him.

The huge amount of material included in the book about Penrose’s parents, his two long marriages and his relationships with his four children seems to me to paint a picture completely typical of his generation. That an upper-class British man growing up in the 1930s and 1940s would have an emotionally withholding father is not very notable. That a male academic of this period would have a marriage that failed after 20 years is not unusual, nor is having a wife with very valid complaints about giving up her own career and interests to follow her husband around to different positions. None of this has anything to do with Penrose’s genius or great accomplishments, beyond the common phenomenon of successful people being too busy and preoccupied to provide enough attention and care to those around them.

The central part of the book is derived from a collection of 1971-76 letters between Penrose and Judith Daniels, a younger woman who had been a childhood friend of his sister. Penrose was unhappy in his marriage, very much in love with Daniels, and saw her as his muse, someone who could appreciate his work. Unfortunately for him, she had a boyfriend and no interest in a sexual relationship or marriage with him. The book goes on for pages and pages quoting these letters and explaining the details of exactly what happened. It’s no more interesting than one would expect. One could argue that Penrose did do something rather objectionable to her, trying to get her to read the manuscript of his two volume joint work with Wolfgang Rindler, Spinors and Space-time.

The four chapters devoted to this story unfortunately are also the ones covering the time of his great work on twistor theory, which gets somewhat buried amidst the not very dramatic unrequited love drama. This section of the book ends with a dubious attempt to connect the two together:

He couldn’t let go of twistor theory, and he couldn’t let go of Judith. In a single letter, he both lamented the impossibility of recreating the magic they once shared and attempted to do exactly that. He wouldn’t take no for an answer — from her or from the universe.

For the years 1971-76, this book provides all the detail you could ever want about why Roger Penrose wanted to sleep with Judith Daniels and why she wasn’t interested. For the details of the story of one of the great breakthroughs in understanding the geometry of the physical world, we’re going to have wait for another book.

Posted in Book Reviews | 32 Comments

Why Sabine Hossenfelder is Just Wrong

Posted on November 11, 2024 by woit

Sabine Hossenfelder’s latest video argues

  1. There’s no reason for nature to be pretty (5:00)
  2. Working on a theory of everything is a mistake because we don’t understand quantum mechanics (8:00).

These are just wrong: nature is both pretty and described by deep mathematics. Furthermore, quantum mechanics can be readily understood in this way.

Actually, the title and first paragraph above are basically just clickbait. Inspired by the class I’m teaching, I wanted to write something to advertise a certain point of view about quantum mechanics, but I figured no one would read it. Picking a fight with her and her 1.5 million subscribers seems like a promising way to deal with that problem. After a while, I’ll change the title to something more appropriate like “Representations of Lie algebras and Quantization”.

To begin with, it’s not often emphasized how classical mechanics (in its Hamiltonian form) is a story about an infinite dimensional Lie algebra. The functions on a phase space $\mathbf R^{2n}$ form a Lie algebra, with Lie bracket the Poisson bracket $\{\cdot,\cdot \}$, which is clearly antisymmetric and satisfies the Jacobi identity. Dirac realized that quantization is just going from the Lie algebra to a unitary representation of it, something that can be done uniquely (Stone-von Neumann) on the nose for the Lie subalgebra of polynomial functions of degree less than or equal to two, but only up to ordering ambiguities for higher degree.

This is both beautiful and easy to understand. As Sabine would say “Read my book” (see chapters 13, 14, and 17 here).

This is canonical quantization, but there’s a beautiful general relation between Lie algebras, phase spaces and quantization. For any Lie algebra $\mathfrak g$, take as your phase space the dual of the Lie algebra $\mathfrak g^*$. Functions on this have a Poisson structure, which comes tautologically from defining it on linear functions as just the Lie bracket of the Lie algebra itself (a linear function on $\mathfrak g^*$ is an element of $\mathfrak g$). This is “classical”, quantization is given by taking the universal enveloping algebra $U(\mathfrak g)$. So, this much more general story is also beautiful and easy to understand. Lie algebras are generalizations of classical phase spaces, with a corresponding non-commutative algebra as their quantization.

The problem with this is that these have a Poisson structure, but one wants something satisfying a non-degeneracy condition, a symplectic structure. Also, the universal enveloping algebra only becomes an algebra of operators on a complex vector space (the state space) when you choose a representation. The answer to both problems is the orbit method. You pick elements of $\mathfrak g^*$ and look at their orbits (“co-adjoint orbits”) under the action of a group $G$ with Lie algebra $\mathfrak g$. On these orbits you have a symplectic structure, so each orbit is a sensible generalized phase space. By the orbit philosophy, these orbits are supposed to each correspond to an irreducible representation under “quantization”. Exactly how this works gets very interesting, and, OK, is not at all a simple story.

Posted in Quantum Mechanics | 20 Comments

Living in a Post-truth World

Posted on November 6, 2024 by woit

I grew up in the 1960s and 70s, at a time when fundamental physics was making huge dramatic progress and Western democracies were changing in equally dramatic ways, mostly for the better. It truly did seem that the Age of Aquarius was upon us, and that human societies were on a consistent route to progress, however uneven. By the late 1970s and early 1980s things had started to change, but that humanity and my chosen field of science were sooner or later moving forward still seemed self-evident.

By the late 1990s the situation started getting more disturbing. The likes of Newt Gingrich started taking over the Republican party, with a highly successful propaganda arm called Fox News running 24 hours a day, pushing lies about the Democrats, especially the Clintons (remember Whitewater?). For some mysterious reason, even the New York Times joined in. In theoretical physics, proponents of a failed theory dominated the subject, putting out endless propaganda to the public such as Michio Kaku’s Hyperspace.

Around this time I started spending a lot of time trying to understand how these things could be happening. If someone is saying obviously untrue things, logically there are only two possibilities: they’re ignorant and believe what they’re saying, or they’re dishonest, know very well that they are lying. Watching this kind of thing for many years, I started to realize that a better way of thinking about what was going on is that for many people (mathematicians being somewhat of an exception) the issue of truth just isn’t very relevant. Newt Gingrich and Michio Kaku likely weren’t thinking at all about whether what they were saying was true, they were thinking about what would get votes, sell books, or otherwise further their goals in life. Gingrich was doing what he was doing to save the republic, Kaku to pursue the dreams of Einstein, but both had enthusiastically entered a post-truth environment.

Over the last decade or two, things have gotten much, much worse. Those with a lot of influence in fundamental theoretical physics have driven the field to intellectual collapse by continuing to heavily promote failed ideas. The scientific method is based on abandoning failed ideas and moving on to better ones. As an undergraduate at Harvard I watched Glashow, Coleman, Weinberg, Witten and others quickly abandoning that which didn’t work and moving on to impressive new ideas, with more of the same at Princeton during my graduate years. These days Harvard Physics features a group of people devoted mainly to propping up the failed string theory program (Vafa with the “swampland”, Strominger with “A+++” and Jafferis with the wormhole publicity stunt). The situation at the IAS/Princeton isn’t a lot better.

On the American democracy front, the Trump phenomenon embodies post-truth in its purest form, with the full triumph now of a movement devoted to saying whatever will get them to power, with less than no interest in whether any of it is true. I’ve spent a lot of time trying to understand why voters in the US voted the way they did in this latest and recent elections. Taking a look at last night’s exit polling, the answer is pretty simple. Rich and poor voted much the same way, but those with the least education voted for Trump by a 28% margin, those with the most education voted for Harris by a 21% margin. The polite term for the first group seems to be “low information voters”, but what’s going on is that education is exactly what gives you the tools to look for the truth and not get taken in by lies.

The situation has gotten dramatically worse in recent years, as people get their information from social media, with the rise of powerful algorithms designed to generate outrage and “engagement” (sometimes designed and funded by bad actors). These send even some of the smartest people around deep into rabbit holes of lies.

So, given all this, how does one live a fulfilling life in a post-truth world? I’m 67 years old, now see little chance I’ll be around to see a return to the sort of world I once knew where what was true mattered. On the citizen in a democracy front, over the last few days I’ve adopted a new policy. When I’m reading anything, at the word “Trump” I stop and move on to something else. Other terms will get added to that algorithm as needed. What’s going on is all too clear, there’s nothing I can do about it, and I need to stop wasting time and energy thinking about it more. I’ve deleted the Twitter account I was using (@peterwoit, not the @notevenwrong blog post announcement account) and won’t anymore waste time in that sewer. I’ll miss stringking42069, but one has to make sacrifices.

On the theoretical physics front, I’ll give up wasting time paying attention to what string theorists are up to, and try to concentrate on more worthwhile intellectual activities. The blog will continue though, since it’s one of the main positive things I can do to make a small dent in the post-truth information environment. I’ve always benefited greatly from the many readers who write to me to tell me about things I may not have seen. Keep those cards and letters coming, especially since I’ll be spending less time looking for something new on the physics side of many of the usual topics I’ve covered.

Due to massive increases in the volume and sophistication of trolling, blog comments are now all moderated. If you want to argue that it’s all the Democrats fault (yes, I know that they have their own post-truth problem with identity politics), or that theoretical physics is doing just fine, please go away. If you have an insightful and constructive suggestion about how to live in the post-truth world, I’m willing to listen.

Update: Violated my own new policy by reading the following two analyses of where we are, which are better informed than my own:

https://newrepublic.com/post/188197/trump-media-information-landscape-fox
https://www.inquirer.com/opinion/commentary/trust-mainstream-media-2024-election-20241110.html

Now shutting off comments and attempting to stick to more productive activity.

Posted in Uncategorized | 58 Comments

The Crisis in String Theory is Worse Than You Think…

Posted on October 31, 2024 by woit

Curt Jaimungal has a piece out, an interview with Lenny Susskind, with the title The Crisis in String Theory is Worse Than You Think…. Some of what Susskind has to say is the same as in his recent podcast with Lawrence Krauss (discussed here). These days, Susskind sometimes sounds like Peter Woit:

We live in the wrong kind of world to be described by string theory. No physicist has ever won a big prize for string theory. I can tell you with absolute certainty that it is not the real world that we live in. So we need to start over.

(interesting that Susskind seems to think the “Breakthrough Prize” is not a “big prize”, maybe because he’s one of the few well-known string theorists who hasn’t gotten one).

Susskind says he himself is working on trying to extend string theory to something different which will work in dS space, not just AdS, but he agrees with my claim that this is something the field has essentially given up on:

I actually don’t know anybody who is working, striving to try to expand the theory into either de Sitter space, which is not supersymmetric, or just more generally into an expanded version of the theory. Older people worked on it in the past. They worked on something called spontaneous breaking of supersymmetry. Don’t worry about what it means. It just means the theory wouldn’t be supersymmetric, and they failed. Now, that’s not a criticism of them. I worked on it, and I failed. That’s not a criticism of anybody, but it’s a fact that there is no precise theory which is not supersymmetric.

That is intolerable, in a sense. It can’t stay that way. We have to describe our world. That’s our purpose, and as I said, I don’t know anybody who’s actually working on that. If you were to send out a message to all the world’s theoretical physicists, anybody working on a generalization of string theory, you’d probably find some yeses, probably mostly among older people, and somehow we have to change this.

I’d argue the field has given up on it because, after decades of work, it’s clear this goes nowhere, and sooner or later Susskind will realize this.

At one point Susskind starts making an odd argument, hard to reconcile with the current state of the subject:

Look, there are still people who believe in the flat Earth, for God’s sakes. There’s people who believe all kinds of weird stuff. Don’t think about individuals. Think about the consensus of the largest fraction of physicists working on these things, and you’ll probably be right. The overall consensus of the field tends to be right. Peculiar individuals, no matter how famous they are, no matter how brilliant they are, if they’re off that consensus, and they’ve been off that consensus for a long time, they’re probably wrong. That doesn’t mean for sure that they’re wrong. Don’t look for the weirdos. Look for what the consensus of the majority of well-respected, highly accomplished physicists believe. And you’ll probably be right. There’s no guarantee of it. There are very few cases where the consensus has gone wrong for a long period of time, just where some offbeat idea of some particular individual suddenly changes everything. I’m not saying it doesn’t happen, but rarely. Penrose, what can I say? He believes all kinds of things that I wouldn’t subscribe to. But more than that, things that the consensus wouldn’t subscribe to.

Besides the weirdo Roger Penrose, he’s no fan of the ideas of another weirdo:

What is Peter Woit? If you look on the Internet, if you look on the archive, he has a small number of papers which are bad. They’re bad mathematics and bad physics. They’re just bad. I probably shouldn’t say that. I probably shouldn’t, but I’m going to say it anyway. He has nothing to offer at all. I assure you that if he had something that was compelling and interesting and that solved some problem, the physics community would notice him. I looked at his papers. I was unimpressed,

I guess his reaction is fair, partly since my own criticism of his work on the landscape is much the same (I have on the other hand had nice things to say about his textbooks, some of which appears as a blurb on the French edition of one of them).
In any case, if you start with the assumption that anything too far off the consensus is going to be unpromising, you don’t need to spend much time looking at my work to confidently evaluate it as having nothing to offer.

Jaimungal does get Susskind to realize that the “if it’s not close to the consensus, it’s probably bad” argument is a dubious one, especially at a time when the consensus research program has clearly failed:

But you’re perfectly right. We should certainly be on the lookout for ideas which are not the consensus. We should be watching for them and not immediately dismiss them because they’re not exactly the same as the ideas that we’ve been pursuing. For sure, we should be doing that. So I would agree with you about that. And maybe we haven’t been diligent enough with some of these ideas…

Most of the people I know, and that might even include myself to some extent, are derisive about a lot of these ideas. And they’re correct that there is a very strong skepticism about them, and maybe to some extent, unfounded. We all know that. There’s nothing hidden about that. The answer is I’ve looked at them, and I don’t find anything compelling about them. If you call that derision, yeah, I am a little bit derisive. However, I would say maybe there are elements in those theories which will come back, come back in some different form, which will connect better with the things which I think are right. And that’s a possibility, which I suspect most of my friends don’t entertain.

Who knows, some day Susskind may come around to the idea that one of the SU(2)s in the 4d Euclidean rotation group being an internal symmetry is not complete nonsense. Once I finish writing up a more detailed version of what I’ve been working on I’ll send him a copy. Maybe I’ll even finally figure out a way to use this to do something new with Kogut-Susskind/Kähler-Dirac versions of fermions, and he’ll be pleased that in 1977 he was on the right track…

Update: Somewhat related to the posting is this new rant from Sabine Hossenfelder. It’s motivated by this from “Professor Dave”, who has 3.4 million Youtube followers and is upset that she is hurting the credibility of scientists by criticizing what has happened in fundamental physics over the past 50 years (a topic he seems to know nothing about). While I disagree with her about some things, I strongly identify with:

Why the fuck is it my fault that cranks think I’m their best friend because I’m pointing out that there’s no progress in the foundations of physics? It’s a fact. We haven’t made progress in theory development for 50 years.

To connect explicitly to the topic of this posting, a big reason for the lack of progress is the way Susskind and other leaders of the field see things. In their minds it’s not possible that the consensus (i.e. groupthink) of GUTs/SUSY/strings of the past 50 years could be wrong. Anyone who argues otherwise is a “weirdo” who doesn’t understand the arguments behind the consensus and can’t possibly have any useful ideas about an alternative.

Posted in Uncategorized | 44 Comments

Various Items

Posted on October 24, 2024 by woit

A few items that may be of interest:

  • Edward Frenkel has a new Youtube show/podcast, entitled AfterMath. I gather that part of the concept here is a follow-on to his book Love and Math, but in this different format. He’s always thought-provoking and well-worth listening to on almost any topic, I’m looking forward to seeing what he does with this.
  • Also in the Bay area, Michael Peskin recently gave a talk on How Should We Think about 10 TeV pCM Colliders?. Much of it is a very sobering look at the possible known ways to build a collider capable of colliding elementary particles at 10 TeV in the center of mass. The technical challenges are daunting and if this is going to get done it’s going to take quite a while and be very expensive.

    Besides the technological and financial problems, he faces up to the main problem of justifying such a project:

    Are the secrets of electroweak symmetry breaking and the Higgs field to be found at 10 TeV ? If we believe in this, we must still find arguments to convince our skeptical scientific colleagues. If we don’t believe in it, we are believing that there is no point in making the next step in collider physics.

    We cannot imagine the future of particle physics without grappling with this question.

    More specifically, he sees the challenge as

    to get the money to build such a collider, we would need

    definitive proof of violation of the Standard Model from HL-LHC or Higgs factories

    or

    a clear and compelling model to be tested (as the MSSM was for LHC).

    In my opinion, this puts a large burden on the theory community

    1. To be sure that an e+e- Higgs factory actually is built
    2. To put forward simple and attractive models of EWSB with a “little hierarchy”

    Unfortunately I don’t see any evidence of any attractive ideas about 2., and the sad history of the hype about SUSY and naturalness means that people are going to be looking a lot more skeptically at any claims by theorists to have such a thing.

  • Speaking of Peskin, if you’re looking for an alternative to Peskin and Schroeder, there’s a recent new QFT book that I’ve seen which appears to be quite good: Introduction to Quantum Field Theory, by Anthony Williams. It doesn’t go as far as Peskin and Schroeder and other textbooks that get seriously into Standard Model physics, but it has a lot more detailed and careful explanations of the basics of relativistic quantum field theory. As such it should be significantly more readable by advanced undergraduates and beginning graduate students.
  • Finally, two questions I’m wondering about, curious if anyone reading this knows the answer:

    Whatever happened about the bet between Ken Lane and David Gross over SUSY? Did Gross pay up?

    What’s going on with the 2025 Breakthrough Prizes? In past years, these things have been announced in September, Hollywood “Oscars of Science” ceremony in the spring. This year, nothing in September, and October is almost over, so wondering if the Breakthrough Prize people have a new concept for the prizes for the coming year.

  • One more thing: I just noticed that the SMF has recently published a 1963 text of Grothendieck’s, his notes for a fall 1963 seminar at Harvard on duality theorems in algebraic geometry. Hartshorne ran the seminar and later wrote up notes, which were published as LNM 20, Residues and Duality.

Update: I’ve confirmed with Ken Lane that it seems David Gross won’t admit that SUSY has been a failure and he’s given up on Gross ever paying off on the bet. For more about the reaction of Gross and others to losing SUSY bets, see here
https://www.math.columbia.edu/~woit/wordpress/?p=8708
For documentation of the 1994 Gross/Lane bet, see page 62 of
https://indico.cern.ch/event/527162/contributions/2159007/attachments/1298122/1936489/deroeck_SUGRA_2016_v4.pdf

Update: Thanks to commenters who have answered both of my questions. Besides finding out what happened with the Gross/Lane bet, this comment explains that the new plan for the Breakthrough Prize is to announce it at the same time it is awarded at the Hollywood “Oscars of Science” event in April (so, 2025 prizes announced and awarded April 2025).

Posted in Uncategorized | 15 Comments

All Langlands, all the time

Posted on October 3, 2024 by woit

Trying to keep track of everything happening in the Langlands program area of mathematics is somewhat of a losing battle, as new ideas and results keep appearing faster than anyone could be expected to follow. Here are various items:

  • Dennis Gaitsgory was here at Columbia yesterday (at Yale the day before). I don’t think either lecture was recorded. Attending his lecture here was quite helpful for me in getting an overview of the results recently proved by him and collaborators and announced as a general proof of the unramified geometric Langlands conjecture. For details, see the papers here, which add up in length to nearly 1000 pages.

    For a popular discussion, see this article at Quanta.

    To put things in a wider context, one might want to take a look at the “What is not done in this paper?” section of the last paper of the five giving the proof. It gives a list of what is still not understood:

    Geometric Langlands with Iwahori ramification.
    Quantum geometric Langlands.
    Local geometric Langlands with wild ramification.
    Global geometric Langlands with wild ramification.
    Restricted geometric Langlands for ℓ-adic sheaves (for curves in positive characteristic).
    Geometric Langlands for Fargues-Fontaine curves.

    Only the last of these touches on the original number field case of Langlands, which is a much larger subject than geometric Langlands.

  • Highly recommended for a general audience are the Curt Jaimungal – Edward Frenkel videos about the Langlands story. The first is here, the second has just appeared here, and there’s a third part in the works. One scary thing about all this is that Frenkel and collaborators are working on an elaboration of geometric Langlands in another direction (“analytic geometric Langlands”), which is yet again something different than what’s in the thousand-page paper.
  • Here at Columbia, Avi Zeff is working his way through the Scholze proposal for a version of real local Langlands as geometric Langlands on the twistor P1, using newly developed techniques involving analytic stacks developed by Clausen and Scholze. This is an archimedean version of the Fargues-Scholze work on local Langlands at non-archimedean primes which uses ideas of geometric Langlands, but on the Fargues-Fontaine curve. Together these provide a geometric Langlands version of the local number field Langlands program, with no corresponding geometric global picture yet known.
  • Keeping up with all of this looks daunting. To make things worse, Scholze just keeps coming up with new ideas that cover wider and wider ground. This semester in Bonn, he’s running a seminar on Berkovich Motives, and Motivic Geometrization of Local Langlands, promising two new papers (“Berkovich motives” and “Geometrization of local Langlands, motivically”), in preparation.

    As a sideline, he’s been working on the “Habiro ring” of a number field, finding there power series that came up in the study of complex Chern-Simons theory and the volume conjecture. According to Scholze:

    My hope was always that this q-deformation of de Rham cohomology should form a bridge between the period rings of p-adic Hodge theory and the period rings of complex Hodge theory. The power series of Garoufalidis–Zagier do have miraculous properties both p-adically and over the complex numbers, seemingly related to the expected geometry in both cases (the Fargues–Fontaine curve, resp. the twistor-P1), and one goal in this course is to understand better what’s going on.

  • Finally, if you want to keep up with the latest, Ahkil Mathew has a Youtube channel of videos of talks run out of Chicago.
Posted in Langlands | 7 Comments

Richard S. Hamilton 1943-2024

Posted on September 29, 2024 by woit

I heard this morning that Richard Hamilton passed away yesterday early this morning. He was a renowned figure in geometric analysis, and a faculty member here at Columbia since 1998. In terms of mortality, the last year or two at the Columbia math department have been grim ones, as we’ve lost five senior faculty at relatively young ages: Igor Krichever at age 72, Henry Pinkham at age 74, Lars Nielsen at age 70, Walter Neumann just last week at age 78, and now Hamilton at age 81.

Richard wrote a short autobiographical piece about himself at the time he was awarded the Shaw Prize in 2011, available here. There’s an interview conducted by his Columbia colleague John Morgan here. Just a couple months ago, Richard was award the Basic Science Lifetime award in Mathematics. You can watch his lecture given in Beijing at the time here.

Richard shared with my four other colleagues that have recently passed away a truly generous outlook on life and other people, very much the opposite of some negative stereotypes of academics as narrow and competitive, hostile to their colleagues and institution. I’ll miss him, as I miss the others we have recently lost.

Update: Frank Calegari has some memories of Walter Neumann here.

Posted in Obituaries | 6 Comments

Is Spacetime Unraveling?

Posted on September 26, 2024 by woit

Quanta magazine has just put out an impressive package of material under the title The Unraveling of Space-Time. Much of it is promoting the “Spacetime is doomed” point of view that influential theorists have been pushing for decades now. A few quick comments about the articles:

  • String theory is barely even mentioned.
  • There is one article giving voice to an opposing point of view, that spacetime may not be doomed, an interview with Latham Boyle.
  • The big problem with the supposedly now conventional view that spacetime needs to be replaced by something more fundamental that is completely different is of course: “replaced with what?”. A lot of attention is given to two general ideas. One is “holography”, the other Arkani-Hamed’s amplitudes program. But these are now very old ideas that show no signs of working as hoped.

    Thirty years ago Lenny Susskind was writing about The World as a Hologram. The idea wasn’t new then and seems to be going nowhere now. It was 17 years ago that Arkani-Hamed started re-orienting his research around the hope that new ways to compute scattering amplitudes would show new foundations for fundamental physics that would replace spacetime. Years of research since then by hundreds of theorists pursuing this have led to lots of new techniques for computing amplitudes (twistors, the amplituhedron, the associahedron, now surfaceology), but none of this shows any signs of giving the hoped for new foundations that would replace spacetime.

Instead of saying any more about this, it seems a good idea to try and lay out a very different point of view which I think has a lot more evidence for it. This point of view starts by noting that our current best fundamental theory has been absurdly successful. There are questions it doesn’t answer so we’d like to do better, but the idea that this is going to happen by throwing the whole thing out and looking for something completely different seems to me completely implausible.

One lesson of the development of our best fundamental theory is that the new ideas that went into it were much the same ideas that mathematicians had been discovering as they worked at things from an independent direction. Our currently fundamental classical notion of spacetime is based on Riemannian geometry, which mathematicians first discovered decades before physicists found out the significance for physics of this geometry. If the new idea is that the concept of a “space” needs to be replaced by something deeper, mathematicians have by now a long history of investigating more and more sophisticated ways of thinking about what a “space” is. That theorists are on the road to a better replacement for “space” would be more plausible if they were going down one of the directions mathematicians have found fruitful, but I don’t see that happening at all.

To get more specific, the basic mathematical constructions that go into the Standard Model (connections, curvature, spinors, the Dirac operator, quantization) involve some of the deepest and most powerful concepts in modern mathematics. Progress should more likely come from a deeper understanding of these than from throwing them all out and starting with crude arguments about holograms, tensor networks, or some such.

To get very specific, we should be looking not at the geometry of arbitrary dimensions, but at the four dimensions that have worked so well, thinking of them in terms of the spinor geometry which is both more fundamental mathematically, and at the center of our successful theory of the world (all matter particles are described by spinors). One should take the success of the formalism of connections and curvature on principal bundles at describing fundamental forces as indicating that this is the right set of fundamental variables for describing the gravitational force. Taking spin into account, the right language for describing four-dimensional geometry is the principal bundle of spin-frames with its spin-connection and vierbein dynamical variables (one should probably think of vectors as the tensor product of more fundamental spinor variables).

What I’m suggesting here isn’t a new point of view, it has motivated a lot of work in the past (e.g. Ashtekar variables). I’m hoping that some new ideas I’m looking into about the relation between the theory in Euclidean and Minkowski signature will help overcome previous roadblocks. Whether this will work as I hope is to be seen, but I think it’s a much more plausible vision than that of any of the doomers.

Update: John Horgan has some commentary here, taking the point of view that discussions of “Beyond Space-time” are fine, as long as you realize what you’re doing is “ironic science” not science.

Posted in Uncategorized | 20 Comments