Things Have Changed

I’ve been busy with other things, but after taking a look today at various new things related to quantum gravity, I was struck by how much things have changed sociologically in that subject over the last few years. Back in the days of the “string wars”, debates about quantum gravity were fiercely polarized. Oversimplifying and caricaturing the situation a bit, the two sides of the quantum gravity debate were:

  • Those interested in loop quantum gravity as well as other more exotic attempts to reformulate the problem of quantum gravity. These people just considered pure quantum gravity and devoted a lot of effort to analyzing the deep conceptual issues that arise. They sometimes considered highly speculative hypotheses, trying out abandoning the usual basic axioms, for instance replacing fundamental axioms of quantum mechanics. Lee Smolin was an influential figure, and the Perimeter Institute a major center for this research.
  • String theorists, who argued that the appearance of spin-two massless mode in the quantized string spectrum showed that string theory was the only way to understand quantum gravity. They claimed that they had a single, very specific and highly technical mathematical structure to study, which obeyed the conventional quantum theory axioms. Their efforts were devoted to specific computations in this theory, and they seemed to regard the other side of the debate as woolly thinkers, caught up in meaningless ill-defined philosophical speculation. The KITP at Santa Barbara, led by David Gross and Joe Polchinski, was a major center for this side of the debate.

These days, things have changed. If you’re at Perimeter, prominent activities include:

On the other hand, it you’re in Santa Barbara these days, you might be participating in a KITP conference on Quantum Gravity Foundations. This is featuring very little about the technical issues in superstring theory being discussed at Perimeter, but a lot of discussion of deep conceptual issues in quantum gravity. There’s also a lot of willingness to throw out standard axioms of physics, maybe even quantum mechanics. They’re even letting Carlo Rovelli talk.

The sort of speculation going on at the KITP is featured on the cover of this month’s Scientific American, and this week Quanta magazine will be publishing a series of pieces on something related, the “ER=EPR” conjecture. There’s debate whether anyone really understands this and whether it is consistent with standard quantum mechanics. It also features a diagram that people call the “octopus” diagram. Back in the day it was Lee Smolin who was getting grief for an “octopus” diagram (see here), yet another way in which things have changed.

For a more balanced view of quantum gravity issues, you might want to spend your time in France, where the IHES recently hosted an interesting series of surveys of the subject (see here), and the Quantum Gravity in Paris conference featured more specialized talks. In the category of quantum gravity topics I wish I had more time to learn about, Kirill Krasnov’s talk was presumably related to this recent work, which looks intriguing.

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The Lady Gaga of French Mathematicians Comes Stateside, and Other News

Cédric Villani is in town today, giving a talk at the French consulate. He’ll discuss his book, recently translated into English (I wrote a bit about it here). Yesterday, despite the lack of suitable bread and cheese, he was in Princeton, where he gave a public lecture at the IAS. The New Yorker has a story about him by Thomas Lin, entitled The Lady Gaga of French Mathematicians Comes Stateside.

If you’re not listening to Villani tonight, you could be watching a PBS Nova program on mathematics, The Great Math Mystery. Among the mathematicians interviewed will be my colleague Dusa McDuff. As for the question on the PBS site:

Is math a human invention or the discovery of the language of the universe?

the answer is the latter.

What some mathematicians might consider the “Great Math Mystery” is whether Mochizuki really has a proof of the abc conjecture. There finally will be the topic of a workshop involving experts in the field, to be held this December in Oxford. Still no paper from Go Yamashita about this, but here you can find some photographs of the boards from his talks in Kyoto last month. Mochizuki himself has a new paper, inspired by conversations with Fesenko.

Also in New York this week, Bjorn Poonen will be speaking on Thursday. His topic is a heuristic argument that there is a finite bound on the rank of elliptic curves. For notes from a talk of his about this last year, see here.

Update: The Villani IAS talk is available here.


Update
: At David Mumford’s blog he has a long and very interesting posting about the state of mathematical research publishing.

Update: One more piece of math news. Dan Rockmore has set up a public version of his Concinnitas Project, which lets people post, with explanation, a picture of their choice of a “most beautiful mathematical expression”. See here for details.

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LHC News

The LHC has just ramped up for the first time to 6.5 TeV, and has a probe beam circulating in one direction, the highest energy protons humans have ever accelerated. You can follow what’s happening here.

The BBC has gotten very excited about this whole LHC thing.


Update
: Now it’s two beams at 6.5 TeV. They just need to be careful to avoid beam collisions until the press event is organized…

Update: Maybe they weren’t careful enough. The Monday morning beam commissioning reports “Possibly first collisions”. No confirmation of this from the experiments, or officially from CERN.

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News of the Multiverse

Just about ten years ago, my April 1 posting here was a fantasy about the Stanford ITP getting major funding from the Templeton Foundation, using it to fund a program on the multiverse, and renaming themselves the Stanford Templeton Research Institute for Nature, God and Science. The last part hasn’t yet come true yet, but I just noticed the announcement last year of a $878K Inflation, the Multiverse, and Holography grant from Templeton to the SITP, the third part of “A three component Templeton Initiative at the Stanford Institute for Theoretical Physics.”

To get some idea of the scale of this funding, note that the entire NSF budget for theoretical HEP is about $12 million (the DOE spends about $50-60 million, but that supports groups at the labs, as well as computational hardware, and is decreasing). The Templeton Foundation has an endowment of over $3 billion (growing rapidly), and pays out over $100 million in grants/year (also growing rapidly). I don’t think my skills as a fantasist are good enough to imagine what this means for ten years from now in the future.

In other multiverse news, the Literary Review of Canada has published a review by David Orrell of the recent Unger/Smolin book, and an exchange of letters between him and Matthew Kleban. I wrote something about the book here, and I’m in many ways not very sympathetic to the point of view of Orrell and Unger/Smolin, especially about the role of mathematics in physics.

I’m more on Kleban’s side about mathematics, but the way he paints multiverse studies as the latest scientific descendant of the mathematics-driven successes of physics of the past is highly problematic. While this is a point of view favored at Stanford and at Templeton (Kleban has a $175,000 grant from them), I don’t think it’s a defensible one. Kleban’s arguments are

  • More to the point is the string landscape, a relatively concrete structure believed to follow from the mathematics of string theory.

    Here “relatively” is a weasel word (relative to what?), masking the fact that we don’t at all know what the structure of the string landscape is.

  • contrary to Unger and Smolin’s assertions, recent work indicates that current or near-future cosmological observations – specifically, the detection of positive spatial curvature – would falsify the landscape (if it is false).

    The situation with the measurement of spatial curvature is that recent Planck results give |Omega_K| less than 0.005 and the expectation is that it is zero to a much higher accuracy than that, way beyond anything measurable (this is considered one of the main arguments for inflation). This “prediction” isn’t “recent”. Susskind’s book on the multiverse ten years ago gave this one bit of sign information as the only prediction of the multiverse (see here). Shortly thereafter some authors were arguing that you could get positive curvature from the string landscape (see here). I have no idea if they’re right, but in a recent paper Kleban himself writes about this:

    Positive curvature would probably not completely end discussion about a multiverse but it would be very bad news for the eternal inflation/CDL bubble nucleation framework.

    and I think Orrell has it right that

    I would be interested to see if the detection of positive spatial curvature actually falsified the theory – wouldn’t it just adapt?

  • Furthermore, the theory can be used to predict the signatures of cosmic bubble collisions: violent events where two previously separate “universes” collide.

    There’s no evidence at all for such “signatures”, and I don’t think there’s any plausible argument for why they’ll appear in new data given that they haven’t been seen yet (I wrote here about Kleban’s Columbia talk about this). Final data from Planck on polarization are expected soon, but this is so implausible that I’m not sure Planck will even bother to look.

    The problem with this kind of “testable prediction” is that it’s much like my claiming that my theory that the universe is controlled by a giant turtle is testable and predictive, since if you saw a big picture of a turtle in the CMB, that would be strong evidence for my theory. There was a reason Popper went on about falsifiability…

  • the standard model of particle physics combined with Einstein’s theory of general relativity – two of the most well-established theories in physics – predict a large landscape quite similar to that of string theory.

    This one brings back the “string wars” era, since I haven’t heard anyone trying to use it (based on this) since 2007. Whenever people make a “string theory is just like the standard model” argument I’m never sure what to respond. How do you argue with someone trying to claim that the most successful physical theory ever, by far, is “quite similar” to a theory that has had zero success? It’s kind of like trying to argue with someone who wants to tell you that black is white, because they’re both kinds of grey. Surely they’re not serious?

    In this case, sure, if you put the standard model on a complicated space-time background, added lots of fluxes, etc. to the background, maybe you could turn it into as useless a theory as string theory. This doesn’t mean it’s “quite similar”.

Update: Just noticed another recent essay about the multiverse, Marcelo Gleiser’s examination of whether Fairies live in the multiverse.

Posted in Multiverse Mania | 35 Comments

Various News

Some news from all over:

  • The problem with a short in the LHC seems to have been resolved (one can follow progress here), looks like they’ll be ready to inject a beam in a few days. Also looks like they’re not likely to spend their Easter Sunday doing this, so, maybe it will be next Monday?
  • Cambridge has finally gotten around to choosing a new Lucasian professor (the last two were Michael Green and Stephen Hawking). Michael Cates will take the position July 1.
  • Grothendieck’s death last year was sad to hear about, but a positive result is that the Grothendieck Circle is back in the business of making available resources concerning his work. There’s a comment at the top of the website that

    With the agreement of Grothendieck’s family, the work of the Circle to bring Grothendieck’s unique story and writings to the public has resumed.

  • There was a workshop this past month devoted to Mochizuki’s work, but I haven’t found anyone who knows what happened there. Minhyong Kim has taken to trying to write about Mochizuki’s ideas on MathOverflow, see here.
  • The Toronto Star has a long article about Langlands.
  • At the KITP this week a new program on quantum gravity is starting. This month’s Scientific American has a Joe Polchinski cover story on Burning Rings of Fire. Maybe some of the KITP talks will be enlightening, but the small amount of time I’ve spent trying to follow the past two years of debate on this has just left me mystified, struggling to see how the very general framework people seem to be working in can possibly lead to a resolution of the questions they’re concerned with.
  • Frank Wilczek has a speculative article about Physics in 100 years. A commenter here suggests comparing it to Wilczek’s version of nearly fifteen years ago. The last fifteen years have not been kind to Wilczek’s hopes for vindication of SUSY or SUSY GUTs, but he’s not giving up yet. It will be interesting to see what his reaction will be if the next fifteen years are equally discouraging.

    I do very much like one thing in the new version, the section about possible unification of ideas of quantization and of symmetry, where he speculates:

    Quantization and fundamental symmetry will not appear as separate principles, but as two aspects of a deeper unity.

    That’s pretty much one of the main motivations of the book I’m writing (see here).

Update: There are rumors going around tonight that there’s been a hoax perpetrated on the arXiv, something like the Sokal hoax. This has to do with an hep-th posting entitled Riding Gravity Away from Doomsday, which has appeared under the name of a very prominent string theorist, Ashoke Sen, winner of the $3 million Milner Fundamental Physics Prize. What I’m hearing is that no one can believe that Sen could possibly have seriously written something this silly, so it must be some sort of hoax. Speculation is that the hoax could have been carried out to make the hep-th moderators look bad, by showing that they’ll agree to anything, no matter how absurd, if it invokes the Landscape and the multiverse. Some think that Sen’s account must have been hacked and then used to post the nonsense paper, others think that Sen himself is behind the hoax, having had enough of the Landscape business. I’ll update this as more information becomes available.

Update: At least some papers on the arXiv still are serious.

Update: Beams are back in the LHC, successfully circulated at 450 GeV on Easter Sunday (live blog here). Next step, ramp up to 6.5 TeV.

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Back from Break

Clouds cleared about 15 minutes too late at Torshavn in the Faroe Islands, so totality was behind a cloud, but still an impressive sight. And the Faroe Islands are quite a remarkable place to visit. Some recent news:

  • The plan has been to inject a beam into the LHC this week, leading to a news item in the UK Daily Express about how Scientists at Large Hadron Collider hope to make contact with PARALLEL UNIVERSE in days. This nonsense comes to us courtesy of this paper published in Physics Letters B.
  • Unfortunately the machine checkout going on at the LHC has identified a problem that may delay contact with the PARALLEL UNIVERSE for a little while. Looks like no beam this week, for details see this from CERN. Some news is put out here, details of discussions of the problem here.
  • Also on the parallel universe front, Quanta magazine has an interview with Weinberg. About the multiverse, he repeats some of the arguments for it, but also says:

    I am not a proponent of the idea that our Big Bang universe is just part of a larger multiverse.

    About string theory, the LHC and SUSY, the exchange went:

    If the LHC finds no evidence for supersymmetry, what happens to string theory?
    Damned if I know!

    Weinberg went on to respond to the issue of the testability of string theory by discussing the possible measurement of primordial B-modes, without mentioning that string theory makes no predictions at all about this.

  • Quanta magazine keeps putting out some of the best coverage of math and physics available. See for instance Natalie Wolchover on penguins (although also read Tommaso Dorigo and Adam Falkowski) and Erica Klarreich on moonshine.
  • Jess Riedel has a wonderful blog posting about the subtleties of the classical limit in quantum mechanics. Textbooks like to claim this is explained by just taking the hbar goes to zero limit of a path integral, but that doesn’t really provide an explanation, for reasons clearly laid out by Riedel.
  • Nominations are open for this year’s Breakthrough Prizes, see here. There will be $3 million prizes in physics and mathematics, as well as $100,000 “New Horizons Prize” for younger researchers, up to 3 each in both math and physics. For more, see here.


Update
: Sabine Hossenfelder performs the public service of reading the “PARALLEL UNIVERSES” paper and explaining what is going on here.

Update: This year’s Abel Prize went to John Nash and Louis Nirenberg. Nature News has a story here. The award to Nash was for his work on PDEs and the Nash embedding theorem. He already has an Economics Nobel, for his work on game theory. This surely makes him the first person to win not-quite-Nobels in two completely different fields.

Update: Also at Nature, news about the LHC problem.

Posted in Multiverse Mania | 33 Comments

Spring Break

I’m heading off soon on spring break, planning on traveling to Scandinavia and hoping to see a solar eclipse. There hasn’t been much news recently from the math and physics worlds, and it’s unlikely I’ll be blogging until I get back (around the 24th), so will turn off comments while away.

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GR100

I hadn’t thought until recently about the fact that this year is the 100th anniversary of Einstein’s discovery of the field equations of general relativity, so there will be quite a few events taking place commemorating this (for a list of some, see here). This week’s Science magazine has a special issue on the topic. It includes news stories about LIGO and gravitational waves, new tests of the equivalence principle, and possible tests of GR from observations of the black hole at the center of our galaxy.

There’s also a review of a book from a few years ago about Einstein’s search for a unified theory, Einstein’s Unification by Jeroen van Dongen. The review addresses something I mentioned in my recent essay about mathematics and physics, that the development of GR provides a good example of a successful theory coming out of not just experiment and “physical intuition”, but motivated also by the serious use of deep mathematical ideas. According to the review:

Einstein employed two strategies in this search [for the GR field equations]: either starting from a mathematically attractive candidate and then checking the physics or starting from a physically sensible candidate and then checking the mathematics. Although Einstein scholars disagree about which of these two strategies brought the decisive breakthrough of November 1915, they all acknowledge that both played an essential role in the work leading up to it. In hindsight, however, Einstein maintained that his success with general relativity had been due solely to the mathematical strategy. It is no coincidence that this is the approach he adopted in his search for a unified field theory.

Besides the fact that Einstein said so, other evidence for the primacy of the mathematical strategy in this case is the simultaneously successful work by mathematician David Hilbert, who was definitely pursuing the mathematical strategy.

While I think there’s an excellent argument that a mathematical approach was crucial in Einstein’s discovery of the field equations, the later history this book deals with also shows the dangers this can lead to. Einstein spent much of the rest of his life on a fruitless attempt to get a unified theory by pursuing the same mathematics he had so much success with in the case of GR. It’s a good idea to keep in mind both examples. On the one hand, trying out some new deep mathematical ideas can lead to success, on a time scale of a few years. On the other, if you’ve spent 30 years pursuing a mathematical framework that has gone nowhere, maybe you should do something else. A lesson that Einstein’s successors at the IAS might want to keep in mind…

The story about new tests of the equivalence principle contains the usual nonsense about testing “string theory predictions”:

Using beryllium and titanium, they found gravitational and inertial mass equal to one part in 10 trillion, as they reported in Physical Review Letters in 2008. That’s not quite precise enough to test string theory predictions.

That “string theory predicts violations of the equivalence principle” is what used to be called a “factoid”, something not true repeated so often that it becomes a fact. It seems though that usage has changed, with “factoid” now often being used to refer to something true. A new word is needed.

Update: See here for an article by Michel Janssen and Jurgen Renn discussing in detail the question of the “mathematical” versus “physical” strategies in Einstein’s discovery of the GR field equations.

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Quick Links

  • The LHC is getting close to the point where it can be restarted with a 6.5 TeV beam energy. Latest news here, schedule here. Plan is for a sector test late next week (beam in part of the machine), beam in the whole machine March 23. First physics run May 18th.
  • Next week there will be a conference in Venice devoted to neutrinos, blogging going on here.
  • There may be some progress on the Mochizuki/abc front. Ivan Fesenko has written up some notes that try and put Mochizuki’s ideas in context with some other more conventional parts of mathematics. The week after next will see a workshop in Kyoto, with lectures from Go Yamashita on the abc proof. Another recent survey talk by Mochizuki is here.
  • The latest AMS Notices has a series of articles about the life and work of Arthur Wightman, one of the main figures in the effort to make rigorous sense of quantum field theory.
  • In my essay about math and physics, I mentioned the Atiyah-Bott work on the moduli space of solutions to the Yang-Mills equation in the case of Riemann surfaces. This has an intriguing analog to the function field case, which was discussed already by Atiyah and Bott. Dennis Gaitsgory has a new paper out that touches on this in the context of his proof (with Jacob Lurie) of the Weil Tamagawa number 1 conjecture for function fields (see here). The new paper has the footnote: “The contents of this paper are joint work with J. Lurie, who chose not to sign it as author.”
  • Returning to physics, Princeton University Press announced that Frank Wilczek will edit a Princeton Companion to Physics, modeled on the wonderful Princeton Companion to Mathematics, which was edited by Tim Gowers. Publication is planned for 2018.
  • Caltech hosted a workshop the past couple days, inaugurating the Walter Burke Institute for Theoretical Physics. Hopefully they’re well-funded enough to put videos or slides online. John Preskill’s remarks at a celebration of the event are available here. He gives some principles for doing science that I very much agree with. In recent years I’ve become especially aware of the importance of his first principle: “We learn by teaching”, since I’ve been learning a lot that way. As the trend grows towards institutes modeled on the IAS and prestigious positions that involve no teaching, I think this needs to be kept in mind.

    I also agree with his last principle: “Nature is subtle”, and found very interesting his comments on the holographic principle:

    Perhaps there is no greater illustration of Nature’s subtlety than what we call the holographic principle. This principle says that, in a sense, all the information that is stored in this room, or any room, is really encoded entirely and with perfect accuracy on the boundary of the room, on its walls, ceiling and floor. Things just don’t seem that way, and if we underestimate the subtlety of Nature we’ll conclude that it can’t possibly be true. But unless our current ideas about the quantum theory of gravity are on the wrong track, it really is true. It’s just that the holographic encoding of information on the boundary of the room is extremely complex and we don’t really understand in detail how to decode it. At least not yet.

    This holographic principle, arguably the deepest idea about physics to emerge in my lifetime, is still mysterious. How can we make progress toward understanding it well enough to explain it to freshmen?

    From what I can tell, the problem is not that it can’t be explained to freshmen, but that it can’t be explained precisely to anyone, since it is very poorly understood. The AdS/CFT conjecture is now older than some of my current students, with a literature of more than 10,000 papers, but taking a look recently (see here) at what should be a toy model case (AdS3/CFT2) reminded me just how little seems to be truly understood. This is a quite odd and I think historically unprecedented situation.

  • Somewhat related to the holography question, for anyone interested in condensed matter physics, I recommend taking a look at Ross McKenzie’s blog Condensed Concepts. He discusses some of the issues related to attempts to use holography in condensed matter. He also has a recent paper (with Nandan Pakhira) showing a violation of a bound suggested by holographic arguments.

Update: On the multiverse mania front, tomorrow Science Friday is hosting Sean Carroll to continue his war against falsifiability and the conventional understanding of science, joined by Seth Lloyd to help promote the multiverse. Perhaps it should be “Pseudo-science Friday”?

Update: Michael Harris’s book now has a blog, which promises to discuss topics that didn’t make it into the book.

Update (March 7)
: Beam is back in the LHC (well, at least in a part of it). There was a successful test today, sending beam into one sector in one direction, two in the other, see here.

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This Week’s Hype

A quarter-century or so ago, one of common arguments for string theory research was that it was “the only game in town”, in the sense that it was the only possible way to get a unified theory. For instance, back in 1987 David Gross had this to say:

So I think the real reason why people have got attracted to it is because there is no other game in town. All other approaches of constructing grand unified theories, which were more conservative to begin with, and only gradually became more and more radical, have failed, and this game hasn’t failed yet.

As years went on and string theory unification went nowhere, this often was replaced by a new “only game in town” argument, that string theory was the only possible quantum theory of gravity. This argument got strong disagreement from people pursuing Loop Quantum Gravity or any number of other ideas.

This week, Quanta magazine has a new version of the argument, reporting that “Researchers are demonstrating that, in certain contexts, string theory is the only consistent theory of quantum gravity. Might this make it true?” The new argument (based on this and this) seems to be that string theory is the only possible theory of quantum gravity because if you look at a certain class of CFTs (based on orbifolds by permutation groups) and invoke the AdS/CFT conjecture for AdS3/CFT2, the 3d gravity theory in the large N limit would have a density of states more characteristic of string theory than a conventional particle theory.

The most obvious problem here is that it is in 3 space-time dimensions, where there are no physical gravitational degrees of freedom. The S-matrix of quantum gravity is exactly calculable in flat 3d space: it’s zero. There’s a very long history of studying 3d quantum gravity, as a toy model without gravitons, but with just topological degrees of freedom. For more about this, see for instance Steve Carlip’s 1998 book on 3d quantum gravity, which works out a large number of different ways of quantizing 3d gravity (not including string theory). One problem with the argument that string theory is the only way to quantize gravity because it is the only way that works in 3d is that, as Carlip shows, there’s a long list of other completely different ways to do this (all arguably not that relevant to the problem since none have gravitons). This is also quite different than the usual argument that string theory is needed to quantize gravity, which is based on the occurrence of a spin 2 graviton in the spectrum of the string theory.

Ignoring the obvious problem of no gravitons and being in the wrong dimension, there are other problems with the argument, for instance the claim that looking at permutation orbifolds tells you about all CFTs, or the claim that a large density of states at high energy means you have to have a string theory. The article quotes Matt Strassler about this:

But these aren’t really proofs; these are arguments. They are calculations, but there are weasel words in certain places… And just finding a stringy density of states — I don’t know if there’s a proof in that … This is just one property.

Carlo Rovelli sums up the issue with using this to hype string theory and excuse its failures:

They should try to solve the problems of their theory, which are many, instead of trying to score points by preaching around that they are ‘the only game in town.’

I haven’t followed closely work on AdS3/CFT2, but it is a quite interesting topic, although not because it promises a proof of the “universality” of string theory. Chern-Simons theory is based on a very similar relation between a topological 3d qft and 2d CFTs, and there we have some idea what is going on, although many fascinating questions remain. One might hope that AdS3/CFT2 provides a context where one could understand things using some ideas from the Chern-Simons context. This is what Witten did back in 2007 in his paper Three-Dimensional Gravity Revisited (I wrote about this before the paper here). My understanding is that problems with Witten’s proposal later surfaced, I’d be curious to hear from an expert on the latest state of that (perhaps Witten can write a “Three-Dimensional Gravity Revisited Revisited” paper).

There are a lot of wonderful questions still not understood about this story, but I don’t see that using it to argue that string theory is the “only game in town” does anything other than throw one more thing on the pile of outrageous hype generated by string theory partisans over the last 30 years.

Update: There’s been a change to the Quanta article, adding to the quote from Lee Smolin, who is making much the same point I was making in this posting:

“And even in that case [2+1 d], there have existed for a long time counterexamples to the string universality conjecture, in the form of completely worked out formulations of quantum gravity which have nothing to do with string theory.” (String theorists argue that these particular 2+1 gravity theories differ from quantum gravity in the real world in an important way.)

This whole thing really is very strange: on the one hand string theorists are arguing that only string theory can give you quantum gravity, based on an argument in 2+1 d. When you point out to them that there are well-known counterexamples to their argument in 2+1 d, they say “well, things are different in 2+1d than in other dimensions”. Just bizarre…

Posted in This Week's Hype | 18 Comments