Not Even Wrong

This week and next there’s an interesting summer school going on at the IAS, with topic Understanding Confinement. Videos of talks are available here or at the IAS video site.

Taking a look at some of the first talks brings back vividly my graduate student years, which were dominated by thinking about this topic. When I arrived in Princeton in 1979, the people there had been working for several years on trying to understand confinement semi-classically, in terms of instantons and other solutions to the Yang-Mills equations (e.g. merons). By 1979 it had become clear that such semi-classical calculations were not sufficient to understand confinement and people were looking for other ideas. There were quite a few around, including the idea that there was some sort of string theory dual to pure Yang-Mills theory, and I spent quite a lot of time reading up on efforts of Migdal, Polyakov and others to find a formulation of string theory that would provide the needed dual. I ended up writing my thesis on lattice gauge theory, an approach which had the great advantage that you could at least put the calculation on a computer and start trying to get a reliable result for pure Yang-Mills numerically. Some of the calculations I did were done at the IAS, with Nati Seiberg and others. The other thing I spent a lot of time thinking about was how to put spinor fields on the lattice, the beginning of my interest in the geometry of spinors.

I strongly recommend watching Witten’s talk on Some Milestones in the Study of Confinement. His career started a few years before mine, with the early part very much dominated by the problem of how to make sense of Yang-Mills theory non-perturbatively, and this has has always been a motivating problem behind much of his work. In his talk he explains clearly the approaches to the problem (lattice gauge theory, 1/N, dual Meissner) that appeared very soon after the advent of QCD in 1973. He emphasizes how each of these approaches shows indications of a possible string theory dual, while frustratingly not leading to a string model that has the right properties, summarizing (41:30) the situation with:

The string theory we want is probably quite unlike any that we actually know, as of now. We don’t know how to make a string theory with the short distance behavior of asymptotic freedom.

In his talk he discusses later developments, in particular the Seiberg-Witten solution to N=2 SYM and the AdS/CFT duality between a string theory and N=4 SYM, explaining how these advances still don’t provide a viable approach to the confinement problem in pure Yang-Mills.

I’m looking forward to seeing the rest of the talks, and finding out more about some things that have happened over the years since I was most actively paying attention to what was happening with the confinement problem.

A few days ago I read a fascinating article in New York magazine: Inside the AI Factory, which explained how the very large business of hiring humans to do tasks that generate training data for AI works. One reaction I had to this was “at least this means math is safe from AI, nobody is going to pay mathematicians to generate this sort of training data.” Yesterday though I ran across this tweet from Kevin Buzzard, which advertises work (see this link) that seems to be of this kind.

A company called Prolific is advertising work paying 20-25 pounds/hour doing tasks in Lean 3. This company is in exactly the business described in the NY mag articles, hiring people to do tasks as part of “studies”, which often are generating AI training data.

One unusual thing about this whole industry is that if you sign up for this work you often have no idea who your employer really is, or what your work will be used for, and you sign a non-disclosure agreement to not discuss what you do. In this case, a few things can be gleaned from discussions on the Lean Zulip server:

• “it’s 40 dollars/hr now actually”
• “I think everyone signed a consent form preventing them from disclosing any problems or even their participation.”
• One problem was formalizing a proof of sin(x)=1 implies x=pi/2 (mod 2pi).
• A representative of the company writes: “Some of our biggest partners are keen to work with Lean users because of its applicability as a theorem prover. They plan to launch numerous studies that require participants to have either a working or expert knowledge of Lean 3.” By “biggest partners”, presumably this is Microsoft/Google or something, not some small publishing organization.

If anyone knows anything about what’s up with this incipient possible math AI factory, please let us know. On the more general issues of math and AI, I’d rather not host a discussion here, partly because I’m pretty ignorant and not very interested in spending time changing that. The situation in mathematics is complicated, but as far as fundamental physics goes, theorem-proving is irrelevant, and applying “big data”/machine-learning/AI techniques to generate more meaningless calculations in a field drowning in them is pretty obviously unhelpful.

For a much better place to read about what is happening in this area, there’s an article in today’s New York Times by Siobhan Roberts: A.I. Is Coming for Mathematics, Too. At Silicon Reckoner, Michael Harris is in the middle of a series of posts about this past month’s workshop on “AI to Assist Mathematical Reasoning.”

No blogging here the past few weeks, partly because I was away on vacation for a little while, but more because there hasn’t been anything I’ve seen worth writing about. Yesterday’s pulsar timing array and IceCube announcements unfortunately didn’t tell us anything about fundamental physics. In the past, such observational results pretty reliably led to absurd claims about evidence for string theory that I could complain about, but that phenomenon seems to be dying down. In this case, the only story that had such claims was one from Quanta Magazine, which explained that “the observations so far from NANOGrav and the other teams are consistent with what we’d expect to see from cosmic strings.”

I noticed that the people at the Institute of Art and Ideas have put together a program for Monday that includes a debate on the topic of “Fantasy, Faith and Physics.” The framing of the debate contrasts the conventional view of science with an alternative possibility: “should we accept that some beliefs, especially in the foundations of physics, are akin to religious beliefs dressed in mathematical language to give our theories meaning?” This kind of misses the point about the current problems in fundamental physics, since I doubt any of the panelists are going to defend such an alternative.

Very odd is what leads into this debate, an interview with Michio Kaku about his new book. Why promote such an atrociously bad book (see here and here) and broadcast Kaku’s absurd claims about this subject?

Maybe this debate will somehow lead to a substantive discussion of the main underlying problem, the nearly fifty-year dominance of a failed set (GUTs/SUSY/strings) of ideas about unification. A very powerful and influential part of the physics community, which will be represented in the debate by Juan Maldacena, continues to insist on the centrality of this set of ideas. To get a clear look at his arguments, see a recent IAI interview In defence of string theory and his colleague Edward Witten’s recent colloquium talk What Every Physicist Should Know About String Theory. The argument Maldacena and Witten are making is essentially the same one from the mid-eighties: string theory is the only possible consistent way to go beyond quantum field theory and get a consistent theory of quantum gravity. In my book and many other places, I’ve explained the many problems with this. Put simply, the problem is that there is no such thing as a well-defined string theory which successfully gives the SM and GR in four dimension. The claims about consistency are either about models that don’t reproduce the real world, or about still-unrealized hopes and dreams (which Penrose characterizes as “Fantasy”) rather than anything well-defined.

For a very clear statement of his point of view from Witten, see the question and answer section of the recent colloquium talk, starting around 1:20, where he starts by emphasizing the rigidity of the framework of relativistic quantum field theory. He then states:

My point of view is that string theory is the only significant idea that has emerged for any modification of the standard framework that makes any sense.

This is pretty much exactly the same argument he was making nearly forty years ago. I didn’t find it convincing then, since it seemed to me there was no reason to be so sure that a deeper understanding of relativistic QFT could not possibly lead to a consistent quantum theory with low energy limit GR. Witten had a good argument in 1984 that a possibly consistent generalization of relativistic QFT was worth studying, but the problem is that decades and tens of thousands of papers later, as far as unification goes, this study has been a failure, taking the field down paths (extra dimensions, SUSY) which lead to complex theories that don’t look at all like the real world.

If you look at where things have ended up and the current research directions Maldacena and Witten are pushing, the odd thing is that they seem to have given up on unification, and for years now have been emphasizing the study of black holes in toy models with little to no connection to string theory. The most disturbing thing I heard in the Witten talk was at 1:24:16

If you had sufficient computing power, maybe with a quantum computer with a million qubits, I think you could simulate the dynamics of a quantum black hole…

Here Witten seems to be pointing to exactly the argument recently made by Juan Maldacena (see here), which has a specific claim about what you could do with a million qubits. This particular calculation would not in any way address the problems of the string theory program and is getting into Michio Kaku/wormhole publicity stunt territory.

Update: There’s an interview with Witten here, associated with his receipt of the Hamburg Prize for Theoretical Physics. About string theory, he explains that, despite 50 years of effort

We don’t understand it very well… In fact, I’d say we only understand a small part. So we’ve been struggling with that ever since the 70s and 80s trying to understand the intellectual framework that it should have been placed in.

He remains convinced though that alternative ideas are not the way to go:

… I find it implausible that physicists would discover a theory that is such a rich source of fruitful ideas about things that are definitely important in other fields by accident. And if we were not on the right track, I would say, it was a big accident. So my personal view is that it would be a cosmic conspiracy if string theory isn’t on the right track.

There is a growing number of critics who complain that string theory is very interesting, but hasn’t really delivered. Because we still have no idea whether it’s correct and we couldn’t make any experiments, which tells us if this is the case. According to you: To what extent is that criticism justified?

Well, not much, honestly. Where critics of string theory have had interesting ideas, they’ve tended to be absorbed as part of string theory. That’s happened several times. Twistor theory, black hole thermodynamics and noncommutative geometry are three examples of interesting ideas. They were by some regarded as alternatives or competitors of string theory, but actually in practice were absorbed as part of the picture in string theory.

This past semester I taught our graduate class on Lie groups and representations, and spent part of the course on the Heisenberg group and the oscillator representation. Since the end of the semester I’ve been trying to clean up and expand this part of my class notes. I’m posting the current version, working title From Quantum Mechanics to Number Theory via the Oscillator Representation. This is still a work-in-progress, but I’ve decided today to step away from it a little while, work on other things, and then come back later perhaps with a clearer perspective on what I’d like to do with these notes. In a few days I’m heading off for a ten-day vacation in northern California, and one thing I don’t want to be thinking about then is things like how to get formulas involving modular forms correct.

There’s nothing really new in these notes, but this is material I’ve always found both fascinating and challenging, so writing it up has clarified things for me, and I hope will be of use to others. The basic relationship between quantum mechanics and representation theory explained here is something that I’ve always felt deserves a lot more attention than it has gotten.

In the past I’ve often made claims about the deep unity of fundamental physics and mathematics, One goal of this document is to lay out precisely one aspect of what I mean when making these claims. There are other much less well understood aspects of this unity, but the topic here is something well-understood.

One thing that struck me when thinking about this and teaching the class is that this is a central topic in representation theory, but one that often doesn’t make it into the textbooks or courses. Typically mathematicians develop theories with an eye to classifying all structures of a given kind. This case is a very unusual example where there is effectively a unique structure. The classification theorem here is that there is basically only one representation, but it is one with an unusually rich structure.

When I get back from vacation, I plan to get back to work on the ideas about twistors and unification that I’m still very excited about, but have set to the side for quite a few months while I was teaching the class and writing these notes. More about that in the next few months…