Not Even Wrong

I was up in Boston for a few days, and managed to attend a few of the talks at the conference in honor of George Lusztig’s 60th birthday. Lusztig started out his career in geometry and topology; his thesis was in the area of index theory, working with Michael Atiyah and using the families version of the index theorem. He soon turned his attention to representation theory, which is the field that he has worked in for most of his career, often from a quite algebraic point of view. His papers are dense and can be difficult to read, especially for someone like me who is not so algebraically inclined, but many speakers at the conference remarked on how their work had drawn important inspiration from one or another of these papers.

Among the things he is famous for are his work on quantum groups, on the representation theory of reductive groups over finite fields (called Deligne-Lusztig theory, for an introduction, see here), on a whole new field in Lie theory known as Kazhdan-Lusztig theory (for an introduction, see the article by Deodhar in the proceedings of the 1991 AMS summer institute on algebraic groups), and many other things.

Of the few talks I heard at the conference, two were really exceptional. One of these was by Michael Atiyah, with the title “Quaternions in Geometry, Analysis and Physics”. He began by explaining that not only was Lusztig 60, but, if he were alive, the Irish mathematician Hamilton would be 200. There’s a famous story about Hamilton’s discovery of the quaternions: this took place in a flash of insight on October 16, 1843, after which he supposedly engraved the defining relations of the quaternion algebra into a Dublin bridge. Atiyah described a piece of history I didn’t know, showing an extract from a 1846 paper of Hamilton’s in which he takes a square root of the Laplacian and essentially writes down the Dirac equation (in Euclidean signature, this was long before special relativity…).

Hamilton was very taken with quaternions as a generalization of complex numbers, and wanted to develop a “quaternionic analysis” that would be a generalization of complex analysis, a project he thought would take him at least ten years. It turns out that you can’t simply generalize the beautiful subject of complex analysis and algebraic geometry over the complex numbers to the quaternionic case. Because of non-commutativity, polynomials behave very differently. Atiyah explained that in his view the correct generalization of complex analysis to the quaternionic case was Penrose’s twistor theory. Here one considers all possible ways of identifying R⁴ with C², forming a 3 complex dimensional “twistor space”. Complex analysis on this twistor space is what Atiyah claimed should be thought of as the quaternionic analog of complex analysis (on the complex plane).

He reviewed the story of how solutions to various linear equations are related to sheaf cohomology groups on the twistor space, then went on to the non-linear case, where solutions of the anti-self-dual Yang-Mills equations correspond to holomorphic bundles on the twistor space. One can generalize twistor theory to what Atiyah claimed should be thought of as quaternionic analogs of Riemann surfaces: 4d Riemannian manifolds with holonomy in Sp(1)=SU(2), these are self-dual Einstein manifolds, what Penrose would call a “non-linear graviton” (although this is the Riemannian, not pseudo-Riemannian case). The twistor space of these 4d manifolds is a 3d complex manifold, and Atiyah considers complex analysis on this to be the quaternionic analog of complex analysis on a Riemann surface.

The quaternionic analog of higher dimensional complex manifolds are manifolds of dimension 4k, with holonomy Sp(k). Unlike in the complex case, there are few compact examples. Atiyah went on to discuss how examples (mostly non-compact) could be generated as quotients using the quaternionic analog of symplectic reduction. He described several different classes of examples, noting that this construction first appeared in work with physicists studying supersymmetric non-linear sigma models. While I was a post-doc at Stony Brook, Nigel Hitchin was visiting there and working with Martin Rocek and others on this, leading to the 1987 paper in CMP by Hitchin, Karlhede, Lindstrom and Rocek. Atiyah said that he wouldn’t try and describe the relation to supersymmetry, since “I don’t know much about supersymmetry, and if I tried to explain it, you would understand even less”. That Atiyah, after many years of working in this area, still finds supersymmetry to be something he can’t quite understand, is an interesting comment, reflecting the way the subject is still very imperfectly integrated into mathematician’s traditional ways of thinking about geometry and algebra.

Atiyah also commented that off and on over the years he had pursued the idea that quantum groups (which aren’t quite groups), are in some sense the quaternionification of a Lie group (which doesn’t quite exist). He said he hadn’t been successful with this idea, but still thought there was something to it, and hoped that someone else would take up the challenge of trying to make sense of it.

The second wonderful talk I heard was that of Igor Frenkel, from Yale, with the title “Quantum deformation, geometrization, categorification: What is next?”. Unlike Atiyah’s talk, which I pretty much completely understood, Frenkel’s covered much too quickly a lot of material I had never understood, but putting it into an intriguing perspective close to the unsolved problems that seem to me the most important ones for mathematicians and physicists to be looking at. Frenkel began by saying that for many years he had been trying to solve the problem of how to generalize the constructions of representations of loop groups that are related to 2d CFT to representations of 3d gauge groups that should be related to 4d QFT. Some of his thoughts about this are in the write up of his talk at the 1986 ICM. He described himself as having for a long time given up on this problem, moving on to simpler things that he could do: quantum groups which are deformations of the affine Lie algebra story. He went on to talk about “Geometrization”, by which he meant the principle that “all structure constants are Euler characteristics of some variety, all vector spaces are cohomologies”, then “Categorification”, to him the principle that “all structure constants are dimensions of vector spaces, all vector spaces the Grothendieck groups of an Abelian category”. Many of the examples he was using to flesh this out are not well-known to me, I need to do some serious work learning about them before I can say that I clearly understand exactly what he has in mind here.

The last part of his talk, the “What Next?”, went by way too fast but sounded fascinating. He claimed to have some new ways of thinking about the problem of what a representation of these higher dimensional analogs of loop groups should me. I hope to learn more from him in the future to get a better idea of what he has in mind here. He and collaborators at Yale have papers forthcoming, which I look forward to reading. When and if I ever better understand this stuff, I may try and write about it again here.

News about various colliders of various vintages:

The Tevatron has been shut down for the past two months for maintenance and various improvements. It should start up again this coming week, for latest news on its status, see here.

Seed magazine is starting a series of articles on the LHC. One prediction about the LHC that I feel confident making is that it is going to get a lot of press coverage.

A group called the LHC Theory Initiative has been trying for a while to get the NSF to fund new postdocs and graduate student fellowships for physicists working on LHC phenomenology. So far they have been turned down, with the panel that recommended not to fund this presumably concerned that money going to this purpose would be taken away from the standard NSF group grants that fund particle theory groups at many institutions. The full NSF proposal is available on-line.

Science magazine has an article about Barry Barish, who is leading the Global Design Effort for the ILC.

The same issue of Science has a paper by Steinhardt and Turok promoting their cyclic cosmological model as explaining the small value of the cosmological constant, together with an article by Vilenkin criticizing them and promoting the anthropic point of view.

Update: Experimentalist Michael Schmitt, sometimes commenter here, has an excellent new blog about accelerator-based particle physics at the Tevatron and LHC entitled Collider Blog.

Update: The Tevatron start-up was going smoothly until early yesterday morning when they ran into serious trouble. Here’s the report from the FNAL accelerator update page:

At 1:24 AM, Operations reported a raccoon attack on the Linac gallery. It seemed to be a coordinated effort. Fortunately, by 1:53 AM, a joint force of operators and Pbar experts managed to drive the raccoons out of their hastily made fortifications. Then at 4:18 AM, the raccoons made what some thought to be a counter attack on the Division Headquarters, but others believed it to be only a simple reconnaissance incursion. No raccoons were either injured or captured during these encounters. Operator losses were low.

Just upgraded to a new version (2.0.2) of WordPress, and as far as I can tell everything is working normally again. If not, let me know. The one change I plan to make here soon is to try out some of the new anti-spam features. The ones in the older version of WordPress were much better at rejecting valid comments than identifying spam, we’ll see about the new version…

Frank Wilczek has a new Reference Frame piece in this month’s Physics Today. It’s about the question of whether the parameters of our fundamental physical theory are uniquely determined by abstract principles, or “environmental”. He gives two reasons for suspicion about the idea that these parameters are calculable from a fundamental theory:

1. They have complicated, “messy” values and, despite much effort, no one has come up with a good idea about how to calculate them (an exception is the ratio of coupling constants in a supersymmetric GUT). He writes:

Could a beautiful, logically complete formulation of physical law yield a unique solution that appears so lopsided and arbitrary? Though not impossible, perhaps it strains credulity.

2. Some of the values are fine-tuned to make complex structures and thus life possible:

It is logically possible that parameters determined uniquely by abstract theoretical principles just happen to exhibit all the apparent fine-tunings required to produce, by a lucky coincidence, a universe containing complex condensed structures. But that, I think, really strains credulity.

Personally I don’t see the same degree of believability problems that Wilczek sees here. On the first point, it seems quite plausible to me that there are some crucial relevant ideas we have been missing, and that knowing them would allow calculation of standard model parameters, by a calculation whose results would have a complicated structure.

On the second, it’s not at all clear to me how to think about this. Sure, the fact that our universe has highly non-generic features means that it is incompatible with generic values of the parameters, but there’s no reason to expect the answer to a calculation of these parameters to be generic. I guess the argument is that there would then be two quite different ways of getting at some of these parameters: imposing the condition of existence of life, and a fundamental calculation; and if two different, independent calculations give the same result one expects them to be related. But the question is tricky: by imposing the condition of the existence of life in various forms, one is smuggling in different amounts of experimental observation. Once one does this, one has a reason for why the fundamental calculation has to come out the way it does: because it is has to reproduce experimental observations.

Wilczek avoids any mention of string theory, instead seeing inflationary cosmology and axion physics as legitmating the idea that standard model parameters are fixed by the dynamics of some scalar fields, or something similar. This dynamics may have lots of different solutions so:

We won’t be able to calculate unique values of the parameters by solving the equations, for the very good reason that the solutions don’t have unique values.

The fundamental issue with any such anthropic or environmental explanation is not that it isn’t a consistent idea that could be true, but whether or not it can be tested and thus made a legitimate part of science. It’s easy to produce all sorts of consistent models of a multiverse in which standard model parameters are determined by some kind of dynamics, but if one can’t ever have experimental access to information about this dynamics other than the resulting observed value of the parameters, why should one believe such a theory? It is in principle possible that the dynamics might come from such a simple, beautiful theory that this could compel belief, but the theories of this kind that I have seen are definitely neither simple nor beautiful. If you want me to believe in a complicated, fairly ugly theory, you need to produce convincing evidence for it, some sort of testable predictions that can be checked. Wilczek does believe that multiverse theories may provide such predictions:

Of course, the very real possibility that we can’t calculate everything in fundamental physics and cosmology doesn’t mean that we won’t be able to calculate anything beyond what the standard models already achieve. It does mean, I think, that the explanatory power of the the equations of a “theory of everything” could be much less than those words portend. To paraphrase Albert Einstein, our theory of the world must be as calculable as possible, but no more.

One can’t argue with this: if a model make distinctive predictions, and these can be compared to the real world and potentially falsify the model, one can accumulate evidence for the model that could be convincing. Unfortunately I haven’t seen any real examples of this so far. The kind of thing I would guess that Wilczek has in mind is his recent calculation with Tegmark and Aguirre that I discussed here. I remain confused about the degree to which their calculation provides any convincing evidence for the model they are discussing.

Unlike many theorists, Wilczek personally seems to be an admirably modest sort of person, and perhaps this has something to do with why the multiverse picture with its inherent thwarting of theorist’s ambitions to be able to explain everything has some appeal for him. Over the years during which particle theory has been dominated by string theory, Wilczek has shown little interest in the subject, perhaps partly due to its immodest ambitions. But I see two sorts of dangers in the way his article ignores the string theory anthropic landscape scenario which is what is driving the interest of much of the theory community in these multiverse models. As his advisor David Gross likes to point out, accepting this scenario is a way of giving up on the perhaps immodest goal he believes theorists have traditionally pursued, and one shouldn’t give up in this way unless one is really forced to. None of these models is anywhere convincing enough to force this kind of giving up.

The second danger is that what is happening now is worse than just giving up on a problem that is too hard. The string theory landscape anthropic scenario is being used to avoid acknowledging the failure of the string theory unification program, and this refusal to admit failure endangers the whole scientific enterprise in this area.

Update: It has been accurately pointed out to me that Wilczek does mention string theory briefly at one point in the article (“Superstring theory goes much further in the same direction”), and alludes to it at another place (when he talks about a “theory of everything”).

Someone wrote in to inform me that Alain Connes has made available at his web-site the full text of his long 1994 book Noncommutative Geometry. This is a rather amazing book, in many ways more of a research document than a purely expository work. All sorts of interesting things in it, mainly about Connes’s ideas linking the “geometry” of non-commutative “spaces” and the theory of operator algebras, much of this via K-theory.

Last week there was a conference on this topic at Vanderbilt. Connes gave talks there focusing on his recent work related to renormalization. Another main topic of the conference was zeta-functions, and recent developments related to Connes’s program for understanding more about them (and perhaps proving the Riemann hypothesis) using ideas from operator algebras and non-commutative geometry. The series of lectures by Consani provide a good introduction to modern ideas about zeta functions and motives that underly this program.

The CERN Council Strategy Group has produced two very interesting “Briefing Books” for its study of strategy for the future of particle physics in Europe.

For something kind of hilarious, see a paper from 2000 pointed out by one of the commenters here. It’s by Gordon Kane, Malcolm Perry and Anna Zytkow and entitled The Beginning of the End of the Anthropic Principle. The authors tell us that in string theory, “in principle, and eventually in practice, all of the masses are calculable, including the up and down quark masses, and the electron mass. There is not any room for anthropic variation of the masses in a string theory.” The opposite conclusion now seems to dominate string theory research, with the paper many people reference as launching the anthropic landscape that of Bousso-Polchinski written a few weeks after Kane et. al. (although Schellekens and no doubt others would claim that they had the idea much earlier).

David Gross recently gave a series of lectures at Princeton entitled “The Search for a Theory of Fundamental Reality” and they are available on-line. When introducing Gross, Curt Callan noted that Princeton University Press hopes that he’ll turn his lectures into a book that they would publish. The last lecture concerns the problems and prospects of string theory and is very similar to one commented on here a couple years ago in the first real posting on this weblog. Gross says about string theory “so far, we haven’t really calculated anything”, and goes on to give three reasons for this:

1. More and more possible compactifications have been found, all of which seem to be equally consistent.

2. Don’t understand how to handle broken supersymmetry.

3. The cosmological constant problem.

The reasons he gives for continued optimism about string theory unification despite these problems are that “we still don’t know what string theory really is”, and there is no consistent picture of cosmology that is understood within the string theory framework.

He explains the anthropic landscape scenario and how it destroys predictivity, then says that some of his colleagues have given up on Einstein’s dream of finding a unique theory with no adjustable parameters, but that he himself won’t do so until he is forced to, and he isn’t forced to yet since we don’t know what string theory is. He made his usual speculation about string theory leading to some still unknown new emergent view of space and maybe time, then went on to give three reasons for supporting continued research in the subject despite its failure to make any progress on its main problems:

1. String theory has given new insights into gauge theory and maybe it will help solve QCD.

2. String theory has given new insights into mathematics.

3. String theory has lead to new speculative phenomenological scenarios (braneworlds).

About point 3. he describes the possibility of evidence for such scenarios showing up at the LHC as “very unlikely” and even says that he is willing to take bets with anyone for any amount of money that the LHC will not see such things (perhaps he should have discussed this with the authors of the recent report that used these scenarios to try and sell the ILC…). I’ve seen this phenomenon before, but it seems to me peculiar to give as a positive argument for string theory that it leads to the study of phenomenological scenarios that you don’t believe.

After his talk, a questioner asked him if string theory might turn out to just be unsuccessful (i.e. wrong), to which Gross responded “String theory can’t be wrong (or even killed)”. He elaborated by saying that it couldn’t be wrong because it was related via AdS/CFT to N=4 supersymmetric Yang-Mills, which was in turn was related to the standard model. Somehow he felt this was an argument that string theory couldn’t be wrong, only incomplete. He acknowledged that recently he had come to the point of view that string theory was not something that led to unique predictions about the world, but that it is incomplete. In this view, string theory is just a framework, like QFT, and some new ideas need to be added to it to turn it into something that really relates to the real world.

Someone asked him about LQG, and he responded by saying that he doesn’t usually comment on LQG in a polite audience, that it wasn’t very successful, didn’t connect to GR, and was not of any interest to physics.

He ended with some pessimistic comments about the possibility that the scientific community might lose the will to go on at some point in the future as it became more and more difficult to get information about shorter and shorter distance scales, or moments closer and closer to the big bang.

Update: There’s an article entitled Hard Landscape by J.R. Minkel in the June 2006 Scientific American. It deals with the Denef-Douglas work showing that finding a vacuum in the landscape with sufficiently small CC is likely to be a computationally intractable NP hard problem.

“The Douglas-Denef paper is surely a problem for drawing conclusions about what the landscape predicts,” asserts Thomas Banks of U.C. Santa Cruz.

Update: Besides the well-known Theoretical Particle Physics Jobs Rumor Mill which deals with tenure-track hiring, there’s now the Theoretical Particle Physics Postdoc Jobs Rumor Mill, which deals with postdocs. This year I count among the postdoc hires 31 string theorists, 12 phenomenologists and 5 hard to characterize, with several major institutions that hire multiple post-docs still only hiring string theorists. The rumors I’ve been hearing that only phenomenologists are getting jobs seem to be complete bunk.

Update: heppostdoc points out that the Postdoc Jobs Rumor Mill is very new and the data is incomplete. Probably complete data would show postdoc hires not as heavily weighted towards string theory.

Update: There’s a conference going on near Washington this week entitled From Quantum to Cosmos: Fundamental Physics Research in Space. Mark Trodden is blogging from the conference over at Cosmic Variance.

Update: Eric Weinstein, who continues to conduct his research in mathematics and physics from within the financial industry here in New York, will be giving a talk at the Perimeter Institute on Wednesday at 2pm, with the title “Gauge Theory of Economics”. Here’s his abstract:

The close relationship between geometry and fundamental physics can be seen from surveying the basic equations underlying the known forces of nature. What has made these repeated appearances of gauge fields and curvature tensors particularly striking in recent years is lack of any comparable applications outside of the Standard Model and General Relativity. In this talk we will pose the question of whether Yang-Mills theory is simply a unifying principle with application well beyond its current use by exhibiting unreasonably effective applications of Gauge Theory beyond those familiar in the Natural Sciences. Armed with these examples, we will then revisit the question about what is most truly special about the Standard Model and Relativity.