Not Even Wrong

In a previous posting, I’d noted that the number of arXiv hep-th preprints/month was significantly higher recently, clearly due to LLM use, but that I hadn’t heard of much in the way of significant new results coming from this. Around the same time, Matt von Hippel issued a challenge to the LLM agents:

So if AI companies want to impress people like me (or scare us, for that matter), then they need to tackle my old field. Show that an AI can take the kinds of computer resources an academic has access to, and solve one of the scattering amplitudes field’s big outstanding problems. Show that a computational limit everyone expected to be a problem doesn’t actually matter. Give us N=8 supergravity to seven loops, or N=4 super Yang-Mills to nine loops.

People at Anthropic took up the challenge, and today announced that Yes, Claude can do Nine Loops. Supposedly this was a “one-shot”:

after asking Claude which problem it was most likely to be able to tackle, they gave it a simple prompt:

“The problem is to compute the Six-particle (hexagon) amplitude in planar N=4 SYM at nine loops.”

From there, they just kept telling it to keep going, with comments like:

“I’m going to sleep and won’t be available for another several hours. Keep working on this until I tell you to stop. Give me updates every 4-6 hours.”

After using one to two thousand dollars worth of Claude time, they got a result. This was sent on September 1 to Lance Dixon, who had techniques that allowed him to validate the result.

The computation of amplitudes like this has become a significant part of hep-th in recent years, heavily promoted by Nima Arkani-Hamed. There’s no practical application of knowing things like this nine loop amplitude, but Arkani-Hamed always has emphasized that the motivation is to use such calculations to get inspiration for a very different fundamental definition of the theory in terms of new geometrical objects. I have no idea whether this nine loop result is helpful for that. It appears that Claude did not come up with a new method, just used known calculational methods. This is an extremely complex sort of calculation.

It will be interesting to see how these new tools change the way those in the large community of theorists doing such calculations pursue their work and publish their results. Will the new AI tools help them come up with new methods for computing amplitudes and new insights into their structure, or will they just make them obsolete, replaceable by any button-presser with a couple thousand dollars and no expertise?

At the other end of hep-th, there’s the even larger community of theorists who, inspired by AdS/CFT, believe the right way to understand quantum gravity is in some sort of quantum information/holographic terms (“it from qubit” is the slogan). Writers at Quanta magazine have always been quite enthusiastic about this direction in hep-th, and today Charlie Wood has a piece about it, entitled Gravity Seems Holographic. What Does That Mean for Reality?. Wood explains the background and latest thinking in this research direction, concluding:

At this point, I subscribe to a more extreme variation of the it from qubit interpretation — mostly just following the rumble of the stampede. I’d bet that the qubits are the real things, but that they don’t live on anything as familiar as a flat screen.

Physicists have tried to stretch the AdS/CFT correspondence to fit de Sitter space — which has no obvious screen — for decades, with limited success. In recent years, they’ve started to get more creative. Susskind and other teams have made progress (opens a new tab) on holographic de Sitter models (opens a new tab) that differ radically from AdS/CFT. Instead of squashing a volume into an area, these universes seem to cram all the dimensions into a lone quantum point. I imagine a bunch of quantum pixels all coexisting in one spot, rather than spreading across a screen. That might be hard to visualize, but we’re already accepting the idea of dropping one dimension of space. Why should tossing the others be so different?

Balasubramanian suspects that even this kind of radical model doesn’t go far enough. Einstein’s theory fused space with time, and so if the three dimensions of space emerge from a spaceless point, then time should emerge from something timeless. Somehow, we and everything we experience exist within an unblinking dot of no size.

I don’t think we’re there yet, but if Claude does become super-intelligent, way beyond our best theorists, we should be able to soon ask: “Claude, is this bullshit or not?” I’m not super-intelligent, but bet that I already know the answer…

Update: More from Matt von Hippel.

As an antidote to the overwhelmingly depressing news of the world, I highly recommend reading the new article in the New York Times about this mathematical result.

There have been some articles about this earlier in places like Scientific American that emphasized the role of Joan Birman, who is still active at 99. I’ve been hearing from Joan regularly over the past few years about this project, and she always described it to me as largely the work of a talented young woman, Vasudha Bharathram. The New York Times story does a great job of explaining the story of this remarkable collaboration.

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.