Not Even Wrong

There’s an interesting new preprint by the historian of mathematics Erhard Scholz about the early history of the use of representation theory in quantum mechanics. Immediately after the beginnings of quantum mechanics in 1925, several people started to realize that the representation theory of the symmetric and rotation groups was a very powerful tool for getting at some of the implications of quantum mechanics for atomic spectra. One of the main figures in this was Eugene Wigner, who was trained as a chemical engineer, but worked on this topic with his fellow Hungarian, the well-known mathematician von Neumann.

Equally important was the role of the mathematician Hermann Weyl, who in 1925 had just completed his main work on the representation theory of compact groups, perhaps the most important mathematical work in a very illustrious career. Weyl was in close communication both with the group at Gottingen (Heisenberg, Born, Jordan) who were developing matrix mechanics, as well as Schrodinger who was working on wave mechanics. Weyl and Schrodinger both were professors in Zurich and knew each other well (Schrodinger’s first paper on quantum mechanics thanks Weyl for explaining to him some of the general properties of equations such as the Schrodinger equation). In 1927/8 Weyl gave a course on quantum mechanics and representation theory, which became the basis of his extremely influential book “The Theory of Groups and Quantum Mechanics”, first published in 1928.

Scholz has also posted another preprint about Weyl’s work, one that focuses on how his conception of the relation between matter and geometry evolved from 1915 to 1930. Weyl worked on general relativity and wrote an influential book about it (Space-Time-Matter, 1918). At that time he, Einstein and others believed that matter could somehow be described by a unified theory expressed in terms of some generalization of Riemannian geometry. Perhaps particles were some specific singularities or special solutions to the non-linear equations for the metric. The advent of quantum mechanics convinced Weyl (unlike Einstein), that this was a misguided notion, that matter should be described by a complex wave function. The right mathematics was not the geometry of a metric, but (in modern language) the geometry of gauge fields and of sections of a vector bundle with connection. The close connection between the basic ideas of representation theory and of quantum mechanics was quite clear to him, so, unlike Einstein, he enthusiastically adopted the new point of view of quantum physics.

One part of the close connection between Weyl and the history of quantum mechanics isn’t mentioned by Scholz. Weyl was not only a close friend of Schrodinger’s, he was Schrodinger’s wife’s lover. Schrodinger didn’t believe much in monogamy; it’s a well-known story that he discovered the Schrodinger equation while on holiday in the mountains with a girlfriend.

A new preprint by Michael Douglas indicates that, at least this week, the latest “predictions” from string theory are for:

1. No large extra dimensions.

2. No low scale supersymmetry.

So it looks like the “prediction” of the string theory “Landscape” will be that no physics related to string theory beyond that of the standard model will ever be observable. Thus the only “prediction” of string theory will be that you can never see any physics related to it. This kind of “prediction” is great since it proves string theory must be true. Either you don’t ever see any effects of string theory in which case you have confirmed its predictions so it must be true, or you do see effects of string theory, in which case string theory is even more true.

Nothing really new at Brian’s physics colloquium today. About 300 people showed up, which I think is probably a record for a physics colloquium. Brian doesn’t like Susskind’s “anthropic” arguments, which shows good sense. He still hopes that some new form of non-perturbative string theory will explain the standard model by picking out the right Calabi-Yau, but admits there’s no known reason for this to happen.

Thomas Larsson wrote in a comment mentioning a news story that appeared early this past summer in the Deseret Morning News (yes, that’s Deseret, not Desert; this is a name Mormons use to refer to Utah). The news story is about a conference on “String Geometry” held at Snowbird, Utah in June. Evidently at Andy Strominger’s talk at this conference someone actually mentioned that there were people who were skeptical about string theory and asked him to comment. His response was that “I hope they’re wrong, but I can’t prove it, and I bet my life work on their being wrong” , which I guess characterizes the attitude of many string theorists these days (“things don’t look good, but I’ve got too much invested in this to give up, so I’ll keep on engaging in wishful thinking even though I no longer have much of an argument for why I’m doing this”).

Many of the talks at the conference are online. These include a couple of interesting talks by Gukov and Spradlin about recent work on twistor theory and perturbative Yang-Mills amplitudes, as well as the usual Michael Douglas talk with its wishful thinking that analyzing the astronomically large “landscape” will somehow lead to some sort of prediction of something. There’s also a talk by Radu Tatar about non-Kahler superstring theory backgrounds. I’ve always wondered about this since I hear from an algebraic geometer colleague that although no one knows whether there are an infinite number of Calabi-Yaus in the Kahler case, if you relax the Kahler condition there definitely are an infinite number of them. If these non-Kahler backgrounds make sense, you can stop worrying about whether the landscape contains 10^100 or 10^500 possibilities.

Tomorrow here at Columbia my colleague Brian Greene is giving a colloquium on “The State of String Theory”. His abstract says he’ll “assess both its current shortcomings and major achievements”.

Lubos Motl has an interesting post on sci.physics.stringsthat gives a detailed explanation of the current state of string field theory.

One way of motivating quantum field theory is to start with a “first-quantized” quantum theory of particles (perhaps defined by integrating over paths), then “second-quantize” by considering a quantum theory of fields, where the fields are defined on the space the points in the path move in. The natural generalization to string theory would be to start with the “first-quantized” theory of strings given by doing path integrals over the possible worldsheets traced out by the moving strings (these are conformal field theories), then “second quantize” by quantizing fields defined on the infinite dimensional space of loops. It has always been a hope of string theorists that this would somehow give a true non-perturbative definition of string theory.

Lubos explains what some of the problems with this idea are. For one thing it is in conflict with the M-theory philosophy that a non-perturbative theory should involve on the same footing not just strings, but also higher dimensional “branes”. He goes on to speculate about what can be done about this problem, saying that perhaps one shouldn’t be trying to find a fundamental set of degrees of freedom and an action functional of them. Instead maybe one just needs to find a set of self-consistent rules, which will be obeyed by all sorts of different degrees of freedom. As he notes at the end, this is similar to the old “Bootstrap Philosophy” of Chew and others that dominated thinking about the strong interactions during the 1960’s. It didn’t work then, and I’ll bet it won’t work now.

This month is the 50th anniversary of the formal founding of the CERN laboratory near Geneva. There’s a very interesting article in Physics World about CERN and its future plans. LHC construction seems to be proceeding more or less on schedule, although there has been a delay in beginning to install the magnets in the tunnel due to problems with the distribution line that will provide liquid helium to the magnets.

Jos Engelen, the chief scientific officer of the lab, is quoted as wanting to see any decision about building a linear collider wait until 2010 or so. The scientific reason for this is that it may take that long to for the LHC to produce results, and the sort of linear collider one wants to build may depend upon these, e.g. on the mass of the Higgs. CERN has its own linear collider technology called “CLIC” which it is working on. CLIC is quite different than the TESLA superconducting cavity technology developed at DESY and recently endorsed by the ITRP committee charged with evaluating which technology to go ahead with. CLIC uses a second electron beam to accelerate the main beam and in principle is capable of higher accelerating gradients than TESLA. Whereas a machine using TESLA technology would probably have an energy of 500 Gev, upgradeable to 1 Tev, CLIC might be able to reach 3-5 Tev. CERN is now increasing the resources devoted to the CLIC project, and clearly hopes that a delay in the decision about whether to build the linear collider would give them time to develop and prove the viability of CLIC.

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.