Not Even Wrong

The latest issue of the Notices of the AMS contains the first part of a long biographical article about Grothendieck written by Allyn Jackson. Evidently Winfried Scharlau is writing a biography of Grothendieck, and Jackson’s article is partially based on materials he has gathered. Much of this material is brought together at a website maintained by the “Grothendieck Circle”.

This issue of the Notices also contains a short expository piece on one of the most abstract ideas due to Grothendieck, that of a “topos”. Illusie was a student of Grothendieck’s, and Jackson’s article has some of his reminiscences about what that experience was like. Illusie’s piece is not very accessible; a better place to try and get some feeling for these ideas is Pierre Cartier’s Bulletin article.

One of the great stories of mathematics in recent years has been the proof of the Poincare conjecture by Grisha Perelman. This has been one of the most famous open problems in mathematics and has been around for about one hundred years. In technical terms the conjecture is that if a space is homotopically equivalent to a three-dimensional sphere it is homeomorphic to the three-sphere. In less technical terms it says that if you have a bounded three-dimensional space in which all loops can be shrunk down to points, it has to be the three-sphere. In dimensions other than three the analog conjecture has been proved, but the case of three dimensions has resisted all attempts to solve it.

Perelman spent time as a visiting mathematician at Stony Brook, Berkeley and NYU, then went back to St. Petersburg where for eight years he seemed to disappear from mathematics research. In November 2002 he posted a preprint on the arXiv, which quickly drew a lot of attention. He seemed to be claiming to have a proof of an even more general conjecture than Poincare, known as the Thurston Geometrization conjecture, but the way his preprint was written, it wasn’t clear whether he was claiming to really have a proof. The method he was using was one pioneered by my Columbia colleague Richard Hamilton, called the “Ricci flow method”. This involves something like a renormalization group flow to a fixed point (for more about this, see the talks by Ioannis Bakas at a recent conference in Crete). If you start with an arbitrary metric on a space you think might be a three-sphere, the hope was that Hamilton’s Ricci flow would take you to the standard metric for the three-sphere. Hamilton had made a lot of progress using his techniques, but as far as pushing them through to give a proof of Poincare, he was stuck.

In the spring of 2003, Perelman traveled to the US and gave talks at several places, including a long series at Stony Brook. By then he was explicitly claiming to have a proof, but few of the details were written down, although he did post two more preprints to the arXiv. His talks were major events in the math community, and at them he was able to answer anyone who asked for details on specific points of his argument. He gave a somewhat informal talk at Columbia one Saturday, a talk that I attended sitting next to Hamilton, who was hearing Perelman speak for the first time. Hamilton was clearly very impressed, and soon thereafter he and most other experts began to become convinced that Perelman really did have a way of proving the conjecture.

By now the situation seems to be that the experts are pretty convinced of the details of Perelman’s proof for the Poincare conjecture. The full Geometrization conjecture requires some more argument and I gather that Perelman is supposed to at some point produce another preprint with more about this. A workshop was held a couple weeks ago about Perelman’s work at Princeton and several people have been carefully working through the details needed to be completely sure the proof works. For this material, see a web-site maintained at Michigan by Bruce Kleiner and John Lott.

One interesting part of this story is that the Poincare conjecture is one that the Clay Mathematics Foundation has put a one-million dollar price tag on. There’s an elaborate set of rules that Perelman should follow to collect his million dollars. This is supposed to begin with the submission of a detailed proof to a well-known refereed journal, something Perelman hasn’t done and shows no signs of doing. As far as anyone can tell, his attitude is that he’s not interested in the million dollars. If you look closely at the rules, it doesn’t necessarily have to be Perelman who writes up the proof. Someone else may do it, with Perelman still getting the money. Ultimately the question of the million dollars is to be decided by the Scientific Advisory Board of the Clay Mathematics Institute, and one question they will have to face is whether to split the award between Perelman and Hamilton.

Another interesting question concerns the Fields medal, the most prestigious award in mathematics. These are awarded every four years at the International Congress of Mathematicians, the next one of which will take place in Madrid in the summer of 2006. One stipulation for the award of the Fields medal is that a recipient must be under the age of 40. Seeing Perelman speak, I had assumed he was already at least forty, but this is not so clear. No one seems to be sure exactly what his age is and whether he will be under 40 in 2006. Some news reports from spring 2003 referred to him as being in his late 30s or even 40, some recent ones claim that he is now 37. His first scientific paper was published in 1985, so he would have had to have been 19 or younger at the time to be under 40 in 2006. If Perelman really is under 38 now, he’s a sure thing for a 2006 Fields medal.

For a really dumb news article about this, go here (no, proving the Riemann hypothesis won’t bring down the internet, and Perelman’s Poincare proof won’t explain the nature of the universe).

When I first started this weblog I thought very few people would be interested in reading it. I’ve been very pleasantly surprised both by the general high quality of the comments people contribute and by the ever increasing number of people reading “Not Even Wrong”. For the last few months I’ve been running a program that gathers statistics from the web server logs. Here are the monthly numbers for accesses to the main weblog page:

May: 4532
June: 7194
July: 8697
August: 10427

These numbers don’t include a lot of the traffic, which consists of people coming directly to one of the postings, via a Google search or a link from somewhere else.

The Division of Particles and Fields of the American Physical Society has been having its annual meeting at UC Riverside during th past few days, and some of the plenary talks have been put on-line.

Particularly interesting is the talk by Jamie Rosenzweig about Advanced Accelerators: Near and Far Future Options. It reviews ongoing development of existing technologies for use in the next (post-LHC) generation of accelerators, including the superconducting RF cavity technology recently chosen for use in a possible electron-positron linear collider. But it also covers some of the more exotic acceleration technologies that people are thinking about, including optical lasers and plasma wake-fields. Some of these technologies, if they could be made to work, hold the promise of creating much higher accelerating gradients and might allow the construction of much higher energy linear accelerators. The future of particle physics may end up depending on the success of these efforts.

The review of New Models of Electroweak Symmetry Breaking is interesting, although mainly in that it shows that the ideas going around about this aren’t very compelling, and perhaps some dramatically new ones are needed. The reviews of heavy flavor and neutrino physics give a good idea of the current experimental situation. Still to be posted are talks by Clifford Johnson on “Current Trends in String Theory” and by Sean Carroll on cosmology. Carroll also has an interesting discussion of the current state of tests of general relativity on his weblog.

A new preprint by Tom Banks is out, about his idea of “Cosmological Supersymmetry Breaking”. One notable aspect of the paper is a new terminology to describe Weinberg’s “prediction” about the cosmological constant. Since the term “anthropic principle” normally applied to this has acquired a bad odor as it becomes clear it is not science, Banks decides to come up with a different name for the argument. He refers to it as the “galactothropic principle of Weinberg”. Let’s see if this catches on…

The latest issue of Scientific American is devoted to articles about Einstein and his legacy. One article in the magazine doesn’t really have much to do with Einstein and I believe would make him gag if he were still around. The article, entitled “The String Theory Landscape” is by Raphael Bousso and Joe Polchinski. In it they claim credit for the pseudo-scientific idea of “explaining” the value of the cosmological constant by the existence of the “landscape” and the anthropic principle. It’s sad to see this nonsense being purveyed by the most respected and well-known popular science publication in the US.

For something more sensible about the anthropic principle, see a recent column from Nature.

Larry Yaffe’s comments about string theory reflect well mainstream opinion in the particle physics community. On matters of fact I think what he has to say is pretty accurate, but I disagree with some of his statements that reflect not facts but scientific judgements. Of his positive comments about string theory, the ones about its impact on mathematics and about AdS/CFT are right on target. For an interesting talk explaining the status of attempts to use AdS/CFT to say something about QCD, see Larry’s colleague Matt Strassler’s talk this month opening a workshop in Santa Barbara (don’t miss the heated exchange at the end of the talk about whether or not this is all just supergravity).

Larry’s comments about “compelling hints” that there is something “deep and meaningful” to string theory and that it has provided “partial insights” into conceptual problems in quantum gravity are hard to to argue with. But while these hints seem to point in the direction of the existence of an interesting 11 dimensional supersymmetric theory, they provide no evidence that it has anything to do with the standard model. Quite the opposite, the evidence of the “landscape” suggests that any attempts to relate such a theory to the real world produce a framework that is completely vacuous, and can never explain anything (or, equivalently, can explain absolutely anything you choose).

The one place where I think I really disagree with Larry is his claim that, indisputably, “string theory is the most promising framework we have for combining quantum mechanics and gravity”. This “most promising framework” locution has been around now for nearly twenty years. It was justifiable when people were just starting to try and understand the implications of superstring theory, but the failure of twenty years of effort by thousands of very talented physicists has to be taken into account. The fact is that despite all this effort, string theorists still don’t have a consistent theory of 4-dimensional quantum gravity and prospects are not promising that this situation is going to change anytime soon.

As part of this “most promising” comment, Larry has critical things to say about loop quantum gravity. I’m no expert on this myself, but, like many theorists, he seems to me to be holding string theory and loop quantum gravity to quite different standards. Lee Smolin recently wrote to me and Larry to respond to Larry’s comments, he allowed me to reproduce his e-mail here:

“Dear Peter and Larry,

Thanks for the comments, most of which I agree with. But in case either of you are interested, Larry’s comments about loop quantum gravity do not reflect the real results.

A side effect of the sociology of string theory seems to be that there is as much ignorance of the genuine results concerning loop quantum gravity and other approaches to quantum gravity as there is overhype in string theory. It is fascinating that, just there are results that are believed to be true in string theory, despite never having been shown, there are results that have been shown, in some cases rigorously, in lqg, about which many people seem not to have heard about, in spite of being published 5-10 years ago on the archive and in the standard journals.

To combat this I wrote a recent review hep-th/0408048 which I would gently suggest reading before making public pronouncements about the status of the field. There are also good reviews on the rigorous side by Ashtekar and Lewandowski and by Thiemann, as well as two textbooks in press from CUP, one from Rovelli and one more rigorous from Thiemann.

Larry says of LQG that it “has not been shown to have anything to do with gravity. Does it have a large-volume limit? Does it have long distance dynamics…”

Can I mention some of the results that show that lqg quite definitely is a quantum theory of gravity, with details and referenceds in the paper? Larry, if you think any of these results are wrong, please tell us on what step of what calculation or proof someone made an error. Otherwise, we invite you to study the results and the methods by which they were gotten. You might surprise yourself by coming to agree with us, after all this is just quantum gauge theory, but in a diffeomorphism invariant setting rather than on a background manifold.

A key result is the LOST uniqueness theorem which shows that for d >=2 the hilbert space LQG is based on is the UNIQUE quantization of a gauge field that carries a unitary rep of the diffeo group, in which both the wilson loop and non-abelian electric flux operator are well defined operators. (see the paper and references for the precise statement).

Given that GR and supergravity are well understood to have configuration spaces defined as configurations of gauge fields mod diffeos, to which the theorem applies, this implies that the hilbert space used is uniquely suited to the quantization of those theories.

It is further shown that the hamiltonian contraint of GR for d=3+1 is rigorously defined on the hilbert space of diffeo constraints, allowing exact solutions to all the quantum constraints to be constructed.

As far as the path integral is concerned, using the method of spin foam models, based on the observation that GR and supergravity in all dimensions are constrained topological field theory, leads to rigorously defined path integral measures corresponding to the quantization of these theories. There are in addition rigorous UV finiteness results. There are also results that establish correspondences with Regge calculus in various limits.

These results all are quite sufficient to establsih that these theories are precisely the quantization of GR or supergravity. Surely this has something to do with gravity.

Regarding the low energy limit, more explicitly, there are several classes of candidate ground states that have the property that 1) measurements of coarse grained geometrical operators agree with classical flat or deSitter spacetime, up to small fluctuations. 2) small excitations of the gravitational degrees of freedom, which satisfy the constraints to linear order in l_Planck/wavelength have two spin two massless degrees of freedom per momentum mode (i.e. the gravitons are recovered for wavelengths long in planck units, again showing this is a quantization of gravity.) 3) after coupling to any standard matter field, excitations of the ground state yield a cutoff version of the quantum matter field theory on the classical background, cutoff at the planck scale because of the finiteness of quantum geometry.

The classes of states these statements characterize are a) coherent states, b) eigenvalues of coarse grained 3-geometry (sometimes called weave states) and c) for non-zero cosmological constant, the Kodama state.

I would think that finding explicit states with these properties proves that at least linearized gravity and effective field theories correspodning to qft on background manifolds is recovered. Certainly this is again something to do with gravity.

In addition, we can mention 1) the black hole results, which give an exact description of the quantum geometry of an horizon, in agreement with all semiclassical results and 2) the loop quantum cosmology results, which again recover all results of semiclassical quantum cosmology and go beyond them in the context of a rigorously defined framework. Again, some things to do with gravity.

See my paper for complete references.

I’m sorry for the tone, but one loses patience after 15 years. We have always been careful to state results precisely with full qualifications and never to overclaim. By now there are sufficient results that I think the many good people who are working hard in this field deserve to have their results much better known.

As always, I and my (rapdily growing number of) colleagues are happy to talk to anyone and go anywhere to explain the results and the methods by which they were gotten. Indeed, the number of invitations for talks at places that previously expressed no interest previously is growing. I’d certainly be glad to recommend good speakers who could educate your department about the state of art in quantum gravity.

Thanks,

Lee

For many years now, the highest priority of experimental particle physicists for a next-generation accelerator project has been a new electron-positron linear accelerator. The last high energy electron-positron collider, LEP, reached a total energy of 209 Gev before being shutdown in 2000. To get to higher energies than LEP, a ring isn’t viable because synchrotron radiation losses go as the fourth power of energy, so a ring with much higher energies than LEP would use an intolerable amount of electric power.

A linear collider, where you build two linear accelerators and collide their beams together, doesn’t have the synchrotron radiation problems (although the electric power demands are still a problem since the beams you accelerate only can collide once, not many times like in a storage ring). There have been several competing designs for a linear collider, with one of the main difference in the designs being whether the RF accelerating structures are superconducting (“cold”) or room temperature (“warm”). These designs all are for a machine that would start out with a total energy of 500 Gev and ultimately reach 1 Tev. A committee was formed called the “International Technology Recommendation Panel” (ITRP), and it has issued a press release announcing its decision today. The ITRP came down on the superconducting side; this is a design mainly developed at DESY in Hamburg as part of the TESLA project.

The German government has decided to use the TESLA technology to construct a free-electron X-ray laser at DESY called XFEL. They have done this in a way that would allow XFEL to ultimately be upgraded to a linear collider at DESY, but have put off any decision about whether to actually fund and build such a machine.

The ITRP decision will allow work on a final design for the linear collider to begin, but the trickiest questions still lie ahead. Where will the thing be built and who is going to pay for it? The order of magnitude of the cost is $5 billion and the general assumption is that this will be an international collaboration. Besides the possibility of siting it at DESY in Germany, sites that have been discussed in the US mainly are at Fermilab in Illinois (being pushed by Fermilab), or somewhere in California (being pushed by SLAC). Even the most optimistic time scales for designing, funding and building a linear collider don’t have it running until late in the next decade. More realistic might be the mid 2020’s. The question of where the machine is located is crucial to the long-term future of the SLAC and Fermilab laboratories. If it is at their site or nearby they have an assured future, if not their future becomes much more problematic. CERN has its own design for an even higher energy linear collider called “CLIC”, but CERN’s funds for the forseeable future are committed to constructing and funding the LHC, as well as possible future upgrades of that machine.

There’s a bewildering array of web-sites with information about this, including the new International Linear Collider Communication and linearcollider.org ones, another one at SLAC, one at Fermilab (which seems kind of out of date..) and Michael Peskin’s home page. This last one contains links to many talks by Peskin about the physics to be done by a linear collider as well as a web page of links to other information about the linear collider.

The ITRP decision was announced at one of the year’s biggest high energy physics conferences, the ICHEP being held now in Beijing. The web site for that conference contains many talks giving the latest results from experimental groups around the world. Unfortunately, as far as I can tell there’s nothing very earth-shattering being reported.

My old friend and Princeton roommate Nathan Myhrvold has written an excellent piece about the anthropic principle and the Smolin-Susskind debate that has just been posted on the Edge web-site. It seems to me to summarize the issues very clearly.

After getting his Ph.D. in quantum gravity at Princeton, Nathan went to work as a post-doc with Stephen Hawking, and one of the topics he worked on involved a possible mechanism for explaining the small size of the cosmological constant. Nathan left physics and joined with some of my other friends from grad school days to start a software company near Berkeley that they called “Dynamical Systems”. They soon sold the company and themselves to Microsoft, where Nathan ended up in the position of Chief Technical Officer. He periodicaly reminds me that if I had taken up one of his many offers to come work with them back in the mid-eighties, I could be obscenely wealthy too. At the time I remember it seemed clear that it was much smarter to stay as a postdoc at Stony Brook, being paid enough to live on and able to think about whatever I wanted, than to go to California to work twenty hour days writing operating system code and getting paid in worthless pieces of paper.

Oh, well.