Not Even Wrong

A few things that may be of interest:

• Survey articles prepared for the 2018 ICM proceedings are starting to appear on the arXiv, and Peter Scholze (who will be getting a Fields Medal in Rio) has put his on his web-site. His title is p-adic Geometry, and it gives an overview of the ground-breaking work he has been doing over the last few years. The last section tells us that
  

  Currently, the author is trying to understand to what extent it might be true that the “universal” cohomology theory is given by a shtuka relative to Spec Z. It seems that this is a very fruitful philosophy.

  For some background about that section, I’d recommend his talk at the 2015 Clay Math conference.

• The New Yorker has a very detailed and interesting profile of Jim Simons and what he is up to with the Flatiron Institute he is now funding here in New York. This new Institute is costing him \$80 million a year, characterized as “a lark” for someone with his assets. David Spergel is running the Center for Computational Astrophysics there, and doing a lot of hiring. When I wrote here about his characterization of multiverse research, his final comment about being able to speak freely because he had tenure left me wondering “wait, what about grants, jobs, etc.?”. From the New Yorker article, I realized that while having tenure may give you some ability to speak freely, having a guy with \$18.5 billion willing to write large checks for you gives you a lot more…
• I’ve just finished teaching a course this semester which concentrated on the formalism for describing geometry in terms of connections and curvature. From the point of view of physicists, this formalism should be of interest because it applies equally well to gauge theory and general relativity. I’d been starting to think again about what light this formalism might shed on how to think about these two subjects together, when last night I noticed a wonderful new article on the arXiv, Gravity and Unification: A review by Krasnov and Percacci.

  This article is an extremely lucid and comprehensive survey of the sort of thing I was thinking about, which can be re-expressed as the question of trying to find, at the classical level, a formalism uniting the vector potentials/field strengths of the SM and the different possible fields used to describe geometry in GR. Some of this has a very long history, going back to the things Einstein was trying in his later years. There have been many different ideas that people have tried since then, and the survey article does a great job of both explaining these ideas, as well as indicating why they haven’t worked out.

  A couple of the general ideas that have always fascinated me make an appearance in the article. One of these is that of what mathematicians call a “Cartan connection”, the idea that you should think of a geometry as locally looking like a quotient space G/H of two Lie groups. A version of this is known to physicists as the MacDowell-Mansouri formulation, which gets a detailed treatment in the article. Another is the idea of using the fact that the complexified orthogonal group in 4 dimensions breaks up into two pieces, sometimes thought of at the Lie algebra level as self-dual vs. anti-self-dual pieces under the Hodge star operation. A version of this idea is known as the Plebanski formulation, and this decomposition is behind the story of Ashtekar variables. These variable have played a crucial role in modern treatments of GR by Hamiltonian methods, as well as the quantization program of loop quantum gravity.

  The focus of the article is on Lagrangian and classical field theory methods for studying these ideas. There’s relatively little about the Hamiltonian story, and also relatively little about the geometry of spinors, two topics that I suspect might provide additional needed insights. For anyone interested in thinking about non-string theory-based ideas about unification of the SM and gravity, there’s a wealth of ideas, references and history here to think about. Perhaps future progress on unification will come from some new breakthrough in this field that shows how to get around the problems identified clearly in this article.

• For surveys of recent work on quantization of gravity and discussion by experts, a good place to look is videos of talks at a recent conference held at the IHES. Videos available here include my fellow Princeton student Costas Bachas surveying the approach growing out of string theory in Holographic Dualities and Quantum Gravity, Carlo Rovelli the opposition in Current Quantum Gravity Theories, Experimental Evidence, Philosophical Implications, and an even-handed overview from Steven Carlip with Why We Need Quantum Gravity and Why We Don’t Have It. Also well-worth watching, both for the talk and the discussion, is Alain Connes on Why Four Dimensions and the Standard Model Coupled to Gravity.

Finally, for fans of Lenny Susskind’s introductory level books on theoretical physics, Andre Cabannes writes to tell me that the most recent volume (which I wrote about here) is being translated by him into French, to appear next year. He also has notes from the lectures on General Relativity, Cosmology, and Statistical Mechanics, for which no book form has yet appeared.

Update: For a detailed account of the event at NYU mentioned here, see this from Jerry Alper.

Back in 2004, the KITP put out a press release (which I wrote about in an early blog post here) announcing that “Newly Devised Test May Confirm Strings as Fundamental Constituent of Matter, Energy”. The press release announced that Polchinski and collaborators had found “the most viable test to date for determining whether string theory is on the right track”, that this test would be performed by LIGO, which “could provide support for string theory within two years.”

This got a lot of attention and was often quoted as evidence that string theory was testable science. In a 2007 article in Physics World, David Gross answers string theory critics with:

String theory is full of qualitative predictions, such as the production of black holes at the LHC or cosmic strings in the sky, and this level of prediction is perfectly acceptable in almost every other field of science,” he says. “It’s only in particle physics that a theory can be thrown out
if the 10th decimal place of a prediction doesn’t agree with experiment.”

LIGO never found any evidence of cosmic strings within two years after 2004, and now the vastly more sensitive Advanced LIGO experiment has just released results of a search. As expected, the results are negative.

Any guess on the probability of a KITP press release announcing that string theory has failed an experimental test? Or of an acknowledgement by Gross that all the “qualitative predictions” of string theory he was using to justify it ten years ago have now all failed, so, by the standard of “every other field of science”, it should be abandoned?

I made the mistake yesterday evening of spending it out in Red Hook, at an event billed as addressing the scientific controversy over string theory. The venue was an arts space called Pioneer Works, the brain-child of artist Dustin Yellin (whose formative early experience with physics is described here). The event was sold out (tickets were free, courtesy of the Simons Foundation), and drew a huge crowd of several hundred, mostly twenty-something Brooklyn hipsters.

The guests brought in to discuss the controversy were David Gross and Clifford Johnson, and the moderator was Janna Levin. Levin began the discussion by asking the two of them where they stood on string theory: pro, con or agnostic? This flustered Gross a bit (he’s one of the world’s most well-known and vigorous proponents of string theory) and Levin somehow took this as meaning that he was agnostic. Finally Gross clarified things by saying something like “I’ve been married to string theory for 50 years, not going to leave her now”.

Things then moved on to the usual well-worn hype about GUTs, string theory and unification. The LHC made a quick appearance, with no mention of falsified string theory “predictions” of supersymmetry. Instead Johnson characterized the discovery of the Higgs as somehow a vindication for this unification program. Gross went on to explain that unfortunately testing string theory requires going to the Planck scale where strings would be obvious, but that this was out of the question with any conceivable technology.

Besides being immune to experimental test, Gross also described string/M-theory as not a theory at all, since we don’t know its equations or principles (according to him, it’s a “framework”, see here). The conversation then degenerated into a long and meandering discussion of the black hole information paradox (to her credit, Levin countered Gross’s claim that string theory successfully explained it by reminding him of Polchinski and the firewall business).

The Q and A session consisted of a series of mostly crackpot questions from the audience. Johnson responded to a woman saying she thought that we were oscillating between two universes by telling her that she could see she was wrong by testing her theory. The sudden appearance of testability as a criterion to shoot down vague ideas surely confused her.

On a positive note, neither Johnson nor Gross were interested in promoting the multiverse, and the audience was spared that.

Johnson has a new book out called The Dialogues, written in graphic novel form. My previous experience with him was a rather unpleasant one more than ten years ago, after the publication of my book. He wrote a long sequence of blog posts about what he called the “Storm in a Teacup”, attacking Smolin and me and our books. Attempts to discuss the issues involved with him in the comment section there were confusing at first, until things finally became clear when he explained that he was refusing to read my book or Smolin’s. Dialogue about science was not something he seemed interested in if it involved uncomfortable criticism of string theory.

His book addresses this controversy with a panel in which the physicist figure explains:

Frankly, that’s mostly driven by the press, and a few attention-seeking individuals. Most people have a more nuanced view… It just does not sell newspapers or books.

On the question of “attention-seeking”, one might want to consult Johnson’s forty-plus long series of blog postings about his participation and appearance in TV and movie programs. As far as books go, in an end-note for this panel Brian Greene’s The Elegant Universe is recommended. After the Q and A, a long line formed for people to hand their credit cards over to an assistant, then get a copy of Johnson’s book and have it signed.

There’s a very intriguing new article out today by Kevin Hartnett at Quanta magazine, entitled Secret Link Uncovered Between Pure Math and Physics (also a video here). It’s about ideas relating number theory and physics from arithmetic geometer Minhyong Kim. He’s evidently on tour talking about them, with two talks on Gauge theory in arithmetic and a colloquium talk on “Gauge theory in geometry and number theory” in Heidelberg, and a talk on Gauge theory in arithmetic geometry in Paris.

In recent years Kim has been working on what he calls “arithmetic Chern-Simons” theory. For details about this, there are papers here, here, here and here, a workshop here, talks here and here. These ideas grew out of a beautiful and well-known analogy between topology and number theory that goes under the name “Arithmetic Topology”. For more about this, see the book Knots and Primes by Morishita, or the course notes by Chao Li and Charmaine Sia.

While these ideas look quite interesting and I have some idea what they’re about, the Quanta story seems to indicate that Kim has something new, an idea about “Diophantine gauge theory” going beyond the arithmetic Chern-Simons business, and with potential applications to deep problems in arithmetic geometry. Unfortunately the mathematical background here is beyond me (you can try to look at Jordan Ellenberg here, and this earlier paper of Kim’s), and as far as I can tell, the only source for details on the conjectured relations to gauge theory is Kim’s recent talks, which aren’t documented anywhere I can see.

I’m sure we’ll be hearing more about this as time goes on. It joins a host of other ideas relating gauge theory and number theory (in the context for instance of the Langlands program), and promises deeper links to come between fundamental ideas about physics and about mathematics.

Update: Some personal background on this story from John Baez.

Update: More about this story here, including a discussion of the use of Kim’s methods to deal with the “Cursed Curve”.

Update: Reddit has a report of the Heidelberg talk here, unfortunately giving just enough information about the talk to make it sound very interesting, too little to figure out what the ideas discussed actually were. For yet more frustration on this front, Kim gave a general talk on the subject last year here with video supposedly available, but no browser I’ve tried can access it. I do hope we’ll soon see slides, notes, a paper, something, anything, so we can figure out what the Quanta article actually was about.

Update: Kim now has a paper out explaining these new ideas: Arithmetic Gauge Theory: A Brief Introduction.

Quanta magazine has an interesting new piece up, an interview of Witten by Natalie Wolchover.

One topic covered in the interview is the question discussed in a recent posting, that of whether a different formulation of QFT exists, one not based on a choice of Lagrangian. Here Witten is non-committal, leaning to the idea such a thing might exist only in special cases:

Now, Nati Seiberg [a theoretical physicist who works down the hall] would possibly tell you that he has faith that there’s a better formulation of quantum field theory that we don’t know about that would make everything clearer. I’m not sure how much you should expect that to exist. That would be a dream, but it might be too much to hope for; I really don’t know…

I find it hard to believe there’s a new formulation that’s universal. I think it’s too much to hope for. I could point to theories where the standard approach really seems inadequate, so at least for those classes of quantum field theories, you could hope for a new formulation. But I really can’t imagine what it would be.

The standard example of where such a formulation might be needed is the 6d superconformal (2,0) theory, about which Witten says:

From the (2,0) theory’s existence and main properties, you can deduce an incredible amount about what happens in lower dimensions. An awful lot of important dualities in four and fewer dimensions follow from this six-dimensional theory and its properties. However, whereas what we know about quantum field theory is normally from quantizing a classical field theory, there’s no reasonable classical starting point of the (2,0) theory.

About the current state of M-theory, there’s this exchange:

You proposed M-theory 22 years ago. What are its prospects today?

Personally, I thought it was extremely clear it existed 22 years ago, but the level of confidence has got to be much higher today because AdS/CFT has given us precise definitions, at least in AdS space-time geometries. I think our understanding of what it is, though, is still very hazy. AdS/CFT and whatever’s come from it is the main new perspective compared to 22 years ago, but I think it’s perfectly possible that AdS/CFT is only one side of a multifaceted story. There might be other equally important facets.

What’s an example of something else we might need?

Maybe a bulk description of the quantum properties of space-time itself, rather than a holographic boundary description. There hasn’t been much progress in a long time in getting a better bulk description. And I think that might be because the answer is of a different kind than anything we’re used to. That would be my guess.

Are you willing to speculate about how it would be different?

I really doubt I can say anything useful. I guess I suspect that there’s an extra layer of abstractness compared to what we’re used to. I tend to think that there isn’t a precise quantum description of space-time — except in the types of situations where we know that there is, such as in AdS space. I tend to think, otherwise, things are a little bit murkier than an exact quantum description. But I can’t say anything useful.

The hope of 22 years ago was that it was non-perturbative string theory which would provide the desired “description of the quantum properties of space-time itself”. Over the years though studies of gauge-gravity duality have moved away from the use of string theory to provide this bulk description. Witten’s take on the current situation: “There hasn’t been much progress in a long time in getting a better bulk description. And I think that might be because the answer is of a different kind than anything we’re used to.” seems reasonable.

It’s interesting to hear that Witten was going back to Wheeler to see if he had any inspiration to offer the current “It from Qubit” program. This requires a patience for the “vague but inspirational” that Witten has more of these days than he used to:

Why do you have more patience for such things now?

I think when I was younger I always thought the next thing I did might be the best thing in my life. But at this point in life I’m less persuaded of that. If I waste a little time reading somebody’s essay, it doesn’t seem that bad.

This patience is not infinite though: among Witten’s many admirable qualities are the way he responds to:

Do you have any ideas about the meaning of existence?

No. [Laughs.]