Not Even Wrong

I’ve been working recently on trying to understand exactly how Wick rotation works in two-dimensional conformal field theory. There is an analog there of some of the issues with Wick rotation of spinors and twistors in four dimensions that I’ve been struggling with.

One of the few times I’ve tried to get help from an AI agent, I asked it to point me to discussions of this topic in the literature. It gave me a bunch of suggestions I already knew about, then emphasized that the two best places were a certain textbook and another set of lecture notes. Since I hadn’t heard of these, this was exciting news, so I immediately went to look them up. Hallucinations, they don’t exist.

In what I have been reading, I’ve repeatedly come up against what has become a pet peeve, related to the second pet peeve discussed here. In the 2d conformal field theory literature, one is continually told that one is dealing with functions of $z$ and $\overline z$, but $z$ is independent of $\overline z$. $\overline z$ appears to mean something that sometimes is the complex conjugate of $z$, sometimes unrelated to $z$. What’s really going on here???

What’s going on is that one needs to complexify something that is already complex. One can do this by thinking of the underlying real object and complexifying that. When you do this, you end up with a pair of complex objects, with a conjugation map that relates them.

If you start with a real vector space $V$, which also happens to come with a complex structure (a notion of how to multiply elements by $i$), then its complexification will be
$$V\otimes_{\mathbf R}\mathbf C=V \oplus \overline V$$
where $\overline V$ is a copy of $V$ with the opposite complex structure, and you have a map
$$\sigma : V\rightarrow \overline V$$
that satisfies $\sigma^2=1$ and is anti-linear
$$\sigma (cv)=\overline c \sigma (v)$$

Things get confusing once one has $V\oplus \overline V$ because one sometimes wants to think of $v\in V$ as the pair $(v,0)$, independent of things of the form $(0,v)$, but sometimes as pairs $(v,\sigma (v))$. The second interpretation is useful, because then $V\subset V\oplus \overline V$ as the set of fixed points of $\sigma$. $\sigma$ is the conjugation map on the complex vector space $V\oplus\overline V$, so its set of fixed points is the real subspace.

In my earlier pet peeve posting, I write about one example of this. Lie algebras in general are real vector spaces, and to study them, one often needs to complexify. Sometime this is straightforward, for instance
$$\mathfrak{sl}(2,\mathbf R)\otimes_{\mathbf R}\mathbf C=\mathfrak{sl}(2,\mathbf C)$$
sometimes a bit less so
$$\mathfrak{su}(2)\otimes_{\mathbf R}\mathbf C=\mathfrak{sl}(2,\mathbf C)$$
(Note that two different “real forms” have the same complexification).

If one wants to study a complex Lie algebra like $\mathfrak{sl}(2,\mathbf C)$, one often has to think of it as a real Lie algebra and complexify, finding
$$\mathfrak{sl}(2,\mathbf C)\otimes_{\mathbf R}\mathbf C=\mathfrak{sl}(2,\mathbf C)\oplus \overline{\mathfrak{sl}(2,\mathbf C)}$$
This comes up in the Wick rotation context, where $\mathfrak{sl}(2,\mathbf C)$ is the Lie algebra of the Lorentz group, and to Wick rotate to Euclidean spacetime, one is supposed to analytically continue in the complexification.

A second context in which this same problem appears is in the quantization of the harmonic oscillator. If one starts with a real oscillator degree of freedom (e.g. a real scalar field mode of momentum $\mathbf p$), quantization using annihilation and creation operators involves identifying the phase space $\mathbf R^2$ with $\mathbf C$ and then complexifying
$$\mathbf C\otimes_\mathbf R\mathbf C=\mathbf C \oplus \overline{\mathbf C}$$
with the first term in the sum corresponding to creation operators, the second to annihilation operators.

But what if you are dealing with a complex scalar field? Then the phase space of a momentum mode is $\mathbf C^2$ and when you complexify
$$ \mathbf C^2\otimes_\mathbf R\mathbf C=\mathbf C^2 \oplus \overline{\mathbf C^2}$$
You now have two kinds of creation operators and two kinds of annihilation operators. One can organize these into annihilation and creation operators for harmonic oscillator states of two kinds, related by conjugation. These are the particle and anti-particle states, which are distinct in the complex scalar field case, identified in the real scalar field case.

Finally getting to 2d conformal field theory, one sees the same kind of issue, but now not for vector spaces but for manifolds. One is looking at real two-dimensional space-time manifolds $\Sigma$ and would like to complexify to do Wick rotation. But it’s useful to start by thinking of $\Sigma$ as a complex manifold, a Riemann surface. So, again one faces the question of how to complexify something already complex.

This starts to get much trickier to make precise than the vector space case, but it does make sense to think of the complexification of the Riemann sphere as
$$S^2\times \overline {S^2}$$
where $\overline {S^2}$ is the Riemann sphere with the opposite complex structure to that of $S^2$. Then one has pairs of coordinates and is now in the situation where people start talking about coordinates $(z,\overline z)$ where $z$ and $\overline z$ are independent.

By the way, this corresponds to a nice two-dimensional analog of the 4d twistor story. There one looks at compactified, complexified spacetime, with the complex conformal group $SL(4,\mathbf C)=Spin(6,\mathbf C)$ acting, with real forms $Spin(4,2)$ in the Minkowski case, $Spin(5,1)$ in the Euclidean case and $Spin(3,3)$ in the split signature case. Here the 2d compactified, complexified spacetime is $S^2\times S^2$, with a global conformal group
$$SL(2,\mathbf C) \times SL(2,\mathbf C)$$
acting, one factor on each $S^2$ factor. The Minkowski real form is $SL(2,\mathbf R)\times SL(2,\mathbf R)$, the Euclidean real form is $SL(2,\mathbf C)$ and it is for this second real form that one is complexifying something complex.

For some more about this, see chapter 3 of Graeme Segal’s manuscript “The definition of conformal field theory”, where he discusses the infinite-dimensional conformal groups that appear, and his remarkable idea that one should take the complexification of $Diff(S^1)$ to be the semigroup of annuli.

The first surprise of this afternoon was finally finding an informed and sensible discussion of the implications of AI agents for hep-th research, in the form of a twitter thread by stringking42069.

The second was learning from the twitter thread about Xi Yin’s ongoing project to have GPT 5.5 write a QFT textbook under his supervision. The past week there had been rumors that he was hired by OpenAI. If he’s now on their payroll, what he’s getting paid to do presumably is this textbook, which is a very active ongoing project.

The current state of the textbook is at this github repository. I don’t see a pdf anywhere there, but you can get the tex files by cloning the repository with

git clone https://github.com/xiyin137/QFT

and then tex’ing

monograph/tex/main.tex

He’s working on this right now, with latest changes 6 minutes ago.  What I downloaded produced a 3527 page pdf document.

I’ve just skimmed through the thing, and it’s quite fascinating, raising a host of questions. This is clearly a work in progress, on its way to a document with tens of thousands of pages (or more…). Also, it’s undoubtedly the first of many such projects to come. I know from experience that writing a textbook is a huge effort, and AI agents very plausibly could take over a lot of the work. So, one set of questions is about what the future of textbooks, specifically QFT textbooks, will be.

The first obvious comment is that this document is useless as a “textbook”, in the sense of something one could use to learn the subject from. No one is going to learn QFT in any useful sense by trying to read these thousands of pages. Sections of it might be useful to experts in the same way that a badly-written research monograph is.

When I wrote a QM (and some elementary QFT) textbook, a big part of the experience was the following process. Starting from a certain conception in my mind of what the right way to think about a topic was, I’d start writing, and then after a while realize that there was a better, clearer way to think about the topic, so lots of material had to be thrown out or completely rewritten. Sometimes I came to this realization because things were getting too complicated and it became clear there was a simpler way. Sometimes the new insight came from getting stuck on a calculation: at one point, days spent chasing signs that wouldn’t match led to understanding that I was thinking about the dual of the vector space I should have been thinking about.

For an AI agent to be able to write a good textbook, I think it will need to somehow embody that kind of process: realizing when a line of exposition needs to be abandoned because there is a better way to describe what is really going on.

For a QFT textbook, a big set of issues is the unsolved problem of what the best way to think about QFT really is. Just going out to the current literature, grabbing what is there and then trying to rework it as a textbook/monograph, results in a huge, undigested mass of various inconsistent ways of thinking. This may be useful for making clearer to us what’s wrong with the current state of the field, but not useful for anyone who wants to better understand what is really going on.

In any case, I’m curious to see how this project evolves, as well as others like it that surely will come. Initially their role will likely be just to provide examples of what doesn’t work: AI agents fed millions of pages of crud will just produce more crud. Can they develop real insight about fundamental issues in theoretical physics, or can human beings with real insight turn them into useful tools for progress? I’ve no idea how that will play out in the long term. In the short term, I think what we’ll see is just more crud, often dressed up and sold to the public as innovation by the PR departments of our new tech oligarchies.

Update: While Xi Yin is showing us what you’ll get if you get your textbooks from AI, other sites like this are showing what you’ll get if you get your news from AI.

Update: An announcement at the QFT textbook project site says that the public version there has been frozen, further development moved somewhere else (unspecified).

Warning: if you follow this blog, you’ve heard this many times before, so can move on to something more interesting now.

There’s a video conversation between Brian Greene and Lenny Susskind from last week here. At 44:02, Susskind has this to say:

One of the chief critics is a colleague of yours, I believe. And he is rather forcefully maintaining that string theory, until it can produce a success of the kind where you actually can produce a number and that number can be checked with experiment, that it doesn’t have any value.

The main reason for this post is just to reiterate that this is not what I think. The problem with string theory as a unified theory is not that it hasn’t made a tested prediction, but that it has made no predictions, of any kind. It’s very clear now that, as a theory of the real world it’s a speculative idea that just doesn’t work. As to whether it has “any value”, you have to first define what “string theory” is. Under some definitions there are things of value, under others not.

Susskind goes on to accurately explain that any well-defined version of string theory (which he calls “String theory”, capital S) definitely doesn’t correspond to the real world. But, he argues, maybe some new, unknown variant of String theory will work. According to him “It’s the only game in town” and “you have to see it through”.

Brian later asks him “Is there anything that you could imagine happening in the field that would convince you that this is time to put it away?” “Finding it mathematically inconsistent” comes up, but he has already said that this is about an unknown new idea that would make things work. “I don’t know the answer to that” is then his answer: nothing would convince him. About other approaches doing better: “I think both you and I probably don’t put a lot of stock in there.”

In case you haven’t seen this, something from a Kurt Vonnegut magazine piece:

A guy with the gambling sickness loses his shirt every night in a poker game. Somebody tells him that the game is crooked, rigged to send him to the poorhouse. And he says, haggardly, “I know, I know. But it’s the only game in town.”

Update: For another think regular followers here won’t find new at all, see this interview with Neil Turok, who shares a lot of my point of view on the state of theoretical physics. I’m also quite sympathetic to the general idea that right-handed neutrinos are behind the dark matter mystery, although don’t know enough about cosmology to know whether the proposal he mentions is promising.

The big AI/math news is the release today of the Leiden Declaration on Artificial Intelligence and Mathematics. It’s an excellent attempt to identify the new threats to the intellectual culture of the mathematics community and begin a discussion of what to do about them. For some discussion, see Siobhan Roberts at the New York Times, and Michael Harris at his substack. There are some endorsements from prominent mathematicians included. From Peter Scholze there’s

This is a wonderful declaration, coming at the right time. The goal of mathematical research is human understanding of mathematics, and so mathematics can only thrive in a community of human mathematicians. It is crucial to preserve this communal spirit. In my experience, mathematical ideas, like children, must be nurtured and grow over the years. Just like I do not want my children to be educated by AI, I am pondering my mathematical ideas without use of AI, and generally avoid reading AI-generated text as best as I can.

and from Kevin Buzzard there’s

Mathematicians should find it quite striking that tech companies are suddenly interested in their work. The Leiden Declaration is a well-thought-through response to what is currently happening, as AI continues to disrupt this space.

The topic of the capabilities of current and future AI agents is something I’m not very well-informed about, but recently I’ve, like Buzzard, been struck by the way in which obscure mathematical work of little if any practical value has been the subject of a massive publicity campaign. Why do most of the people I meet seem to want to talk about how AI has solved some Erdos problems?

Reading the newspaper financial pages I think provides the answer to this. In today’s New York Times or Wall Street Journal, you can read about the upcoming trillion-dollar IPOs of OpenAI and Anthropic. Hundreds and hundreds of billions of dollars are riding on the relative perception of the technologies of these two companies, something that goes a long way toward explaining why they are throwing massive resources at proving Erdos conjectures and publicizing their successes.

Part of this story is the hiring of prominent mathematicians and physicists to work for very large salaries (and a piece of the trillion-dollar IPO payout). The huge US economic inequality disaster is about to get a lot worse and it makes sense that those with the opportunity to do so would sign up to be on the winning side with the new class of oligarchs. Like most things these days, I’m not sure what’s true and what’s not, but there’s reporting yesterday that Harvard string theory Xi Yin has joined OpenAI. Stringkin42069 is now back, with some pungent commentary on the situation.

String theory and its role in the destruction of a crucial part of our scientific culture is now a story not about the future but about the past. In its post-scientific phase, you can now read about String Theory for Metaphysicians, which is basically a rehashing of the highlights of decades of hype, carefully ignoring the existence of any critique of the hype.

Here at Columbia it’s a beautiful day, and once you get through Checkpoint Charlie, the campus is as attractive as ever. The army of security guards has done an excellent job of keeping us “safe”, which basically means no publicly visible criticism of the state of Israel and the genocide in Gaza. The US/Israeli wars and threats to end Iranian civilization with massive bombardment have drawn attention away from the ongoing genocide in Gaza and ethnic cleansing of the West Bank. Looking back, the Columbia students with their encampment were very much right about what was going on. You can read more about the current situation in Gaza here or here. Netanyahu’s announced intention to kill enough Palestinians to get them out of 70% of Gaza (with 100% for later) has received little attention.

Part of the end of our civilization is that we now live in a post-truth environment, with a Fascist dictatorship in power and oligarchs doing their best to exploit this to their own ends. Finding out what is really happening in Gaza or elsewhere is not easily done. In the Israeli genocide story, one thing that has clarified things for me is listening to people I know well like Scott Aaronson, who with his characteristic clarity of thought explains the logic of why Palestinians and their children must be killed (they want to kill his family). It’s all too clear here that what’s going on is a descent from civilization into tribalism, with Scott’s latest explaining that he will cut off all ties with one of the few relatively reliable sources of information around (the New York Times), because it is telling him things he doesn’t want to hear.

To try and end on a more positive note, I recently found out that Edward Witten is on bluesky, with posts like this that provide some counterpoint to the Scotts of the world. Once you get out of Columbia’s gates, you’re in the city of New York, which now has a wonderfully talented and sensible mayor, who refused to join Israeli officials like Bezalel Smotrich as they marched through Manhattan on Sunday. His statement was:

You can see in the participation of the far-right Israeli Minister Smotrich, as well as a number of other ministers, a vision of annihilation, a complicity in genocide and frankly, a belief that does not have much value for even the sanctity of children in Gaza… And I am offended, as I know many New Yorkers are, by their participation.

Maybe there’s still hope…

Update: Weird thing is that I’ve finally seen an intelligent, reasoned discussion of the implications of AI agents for hep-th research, in the form of a twitter thread by stringking42069.

This posting is mainly intended to provide some links to material about AI in math and physics that I’ve found interesting. I confess that to a large degree I’m trying to avoid seriously learning about exactly what is going on, so my own opinions and thoughts about this topic aren’t grounded in any expertise. If you have interesting things about this topic to point to in the comments, please do so, but for general discussion, try some other venue managed by someone better informed.

• A Future of Mathematics event just started a few minutes ago at Stanford. You’ll find a link to the livestream there and talks should be on Youtube.
• You can easily find people announcing that AI is about to make mathematicians obsolete. We’ll see. In the meantime what I’ve found interesting is that AI is motivating deeper thinking about what what it is that mathematicians really do, and how to protect the valuable parts of this. For good examples of this, see the substacks of Michael Harris and David Bessis. I especially like this recent posting. Also, it was from Michael Harris I learned that Peter Scholze has publicly expressed the opinion that
  

  I already consider the influence of AI to be strongly negative, for humanity, for democracy, and for the planet.

• One of the main problems with AI agents in general is that they are better at saying things that are convincing than they are at saying things that are true. Their potential application in mathematics has the big advantage over other fields that one can use these agents together with formalization and proof-verification to deal with this problem. Scholze has been involved in a major effort using proof verification and I don’t think his remarks about AI apply to this. For a very interesting recent interview with him, see here, which includes some comments about why he hasn’t found formalization that useful.

  Something useful that may come out of this is a conclusive demonstration that there’s a gap in the Mochizuki abc proof. There’s a project working on formalizing this proof announced here. From what I can tell, the situation so far is that the very few who think Mochizuki has a proof have been unable to explain to anyone else how the proof is supposed to work at the point where Scholze/Stix pointed to a gap, and this includes the people charged with trying to formalize this part of the proof.

• In fundamental theoretical physics, formalization is generally not relevant (except perhaps in some areas that could be described as mathematical physics). Given the fact that the subject has been stuck for a long time, with a lot of research devoted to ever-more irrelevant calculations, it seems clear that AI agents likely will soon be able to do this better than humans. For an example of what I mean, see here.

  There is a huge amount of money being thrown in this direction. As an example, the DOE is promoting a Genesis Mission. I’ve no idea how fruitful this will be for most of its goals, but the one relevant to fundamental theoretical physics is “Unifying Physics from Quarks to the Cosmos”. The idea is that

  An AI that internalizes the Standard Model could accelerate analysis by orders of magnitude, identify anomalies pointing to new physics, and propose theoretical extensions consistent with all data—a leap from pattern matching to physics reasoning.

  which doesn’t look at all promising.

  Jared Kaplan tells us here that in 2-3 years AI agents will be replacing the best of IAS theorists. Seems unlikely to me, but we’ll see soon…

Update: Some interesting discussion of this on X (kind of weird to see this in the middle of the intellectual sewer X now is…). See this by Jacob Tsimerman, this by Daniel Litt

Update: Timothy Gowers reports on his latest experience with an AI agent and implications for mathematics research. A team at Google DeepMind has produced an “AI Co-Mathematician“.

Update: Kirti Joshi has made available here his concerns and comments concerning the project to formalize the Mochizuki proof.