Not Even Wrong

The second talk I heard yesterday at the Institute was by Chris Woodward from Rutgers. What he was talking about was a conjectural formula whose origins go back to a truly amazing paper by Witten from 1992 entitled Two Dimensional Gauge Theories Revisited.

There are quite a few very interesting things about this paper, but one of its ideas has become influential in mathematics under the name “Witten localization”. This involves a new principle for calculating integrals of equivariant cohomology classes. Before Witten’s work, it was well-known among mathematicians that such calculations could in many cases be reduced to a “localized” calculation about the fixed point set of the group action. This is related to the Atiyah-Bott version of the Lefschetz fixed point theorem they discovered in the mid-sixties, to general arguments about equivariant K-theory and fixed points due to Atiyah and Segal, as well as to the Duistermaat-Heckman theorem and generalizations due to Berline and Vergne. For some expositions of this material, see the paper by Atiyah and Bott published in Topology in 1984, and the book Heat Kernels and Dirac Operators by Nicole Berline, Ezra Getzler and Michele Vergne.

Witten’s idea involved a new localization principle, where integrals of equivariant cohomology classes can be localized about zeroes of the moment map rather than fixed points of the group action. This is sometimes referred to as “non-abelian localization” since it applies directly to non-abelian group actions, whereas the earlier fixed point formulas typically looked at the fixed points of actions by abelian groups.

One of the main applications of Witten localization by mathematicians has been to use it to prove in various contexts that “quantization commutes with reduction”. For physicists this is the idea that, given a classical mechanical system with a gauge symmetry, one hopes to get the same result either by first imposing constraints and then quantizing, or by quantizing and then imposing constraints. Even in the context of finite-dimensional classical mechanical systems, that this should be true is a very non-trivial mathematical statement. For a survey of some of this, see an article in the Bulletin of the AMS by Reyer Sjamaar.

Witten’s original paper applied his ideas to the calculation of the Yang-Mills partition function in two dimensions. This uses the fact that the space of connections for a non-abelian gauge theory in two dimensions is an infinite dimensional symplectic manifold, with moment map the curvature of the connection, something first observed by Atiyah and Bott in the late seventies.

Woodward’s talk involved an integration formula similar to Witten’s original one, for details about it see his recent paper. This kind of formula was also studied by Paul-Emile Paradan, see this paper and a recent detailed summary (in French) by Paradan of his work.

Woodward’s work is also motivated by trying to understand the 2-d Yang-Mills partition function as an integral in equivariant cohomology. He and Constantin Teleman have done work on a K-theoretic version of this, see their joint paper as well as Teleman’s contribution to the proceedings of the conference in honor of Graeme Segal’s 60th birthday, and his talk at the KITP in Santa Barbara last year.

I was down in Princeton at the Institute yesterday and heard two interesting talks. The first was the beginning of a series of four lectures by Paul Baum about the Baum-Connes conjecture, the second was by Chris Woodward about “equivariant localization”. I’ll write a bit about Baum-Connes here, perhaps something about the topic of Woodward’s talk in a second posting.

The Baum-Connes conjecture was first formulated in 1982 by Paul Baum and Alain Connes in an unpublished paper.Roughly it says that the K-theory of the reduced C* algebra of a group G is identical with the equivariant K-homology of a certain sort of classifying space for the group. Equivariant K-homology classes can be represented by certain generalizations of the Dirac operator, and the map to the K-theory of the C* algebra is given by taking the index of the operator.

There’s a huge literature about this by now and a few years ago Nigel Higson put together a detailed bibliography. Some recent expository articles about the conjecture include an ICM talk by Higson, a survey talk by Wolfgang Lueck, and a book by Alain Valette.

The conjecture remains unproved for discrete groups in general, and Baum said that he suspects it is not true in full generality, invoking what he called “Gromov’s principle”. According to Baum, this principle states that “No statement about all finitely presented groups is both non-trivial and true.” While the conjecture has been proved for some classes of discrete groups, there are many for which it is expected to be true but remains unproved (e.g. SL(3,Z)). For a while it was thought that the conjecture applied also to groupoids, but counterexamples for groupoids have been found.

I’ve always been fascinated by part of the philosophy behind the Baum-Connes conjecture, which is to use equivariant K-homology, classifying spaces and Dirac operators to get information about representation theory of groups in cases where little is known about this representation theory. This is in some very vague sense related to what seems to me to be going on in QFT, where Dirac operators and path integrals over the classifying space of the gauge group (the space of connections) are somehow related to the representation theory of the gauge group. The Baum-Connes conjecture itself just involves locally compact groups and thus doesn’t say anything about gauge groups, so it is not directly relevant to the case that may be of interest in QFT.

An excellent review article about the state of the proof of the Poincare conjecture by my colleague John Morgan has recently appeared. For more background on this, see an earlier posting. Morgan is a topologist, and his article contains an excellent survey of what this all has to say about the topology of three-manifolds. This past semester he has been teaching a course in which he has gone through Perelman’s proof very carefully. So far it all holds together.

The U.S. Congress has finally gotten around to producing a budget for fiscal year 2005. Some information about the budget numbers for scientific research is available here and here.

The NSF budget for research and related activities is being cut by .7% from its FY 2004 level, the first such cut in many years. The other main part of the NSF budget, that devoted to education, is being cut even more. A few years ago Congress passed a bill that was supposed to double the NSF budget over several years, but that bill is now very much no longer operative. It’s not clear yet how physics and math specifically fare under this new budget, presumably we’ll find out in the next few days.

The bulk of particle physics funding comes from the DOE Office of Science, and there the budget situation is brighter, with an increase of 2.8% for FY 2005. Again, the details of exactly what is being funded and what isn’t should soon be available.

Update: More about the new NSF and DOE budgets can be found here and here.

For a depressing look at where theoretical physics is headed, see this new article from Time magazine. I agree with the analysis of it posted here.

In the comment section of the last post, Lubos Motl points to the story of Billy Cottrell, a young string theorist at Caltech accused of being involved in the vandalism of SUVs. Evidently he has now testified against others at his trial, so the “Free Billy Support Network” (which was asking people to send string theory papers to him in prison) has been disbanded and he is being referred to as a “police informant”.

Despite being a string theorist, Cottrell seems to not be the brightest bulb around, having supposedly used a Caltech computer he was logged into to send an anonymous e-mail to the media claiming responsibility for the SUV vandalism.

The local Pasadena newspaper’s report on his testimony at his trial says that he corrected Judge Gary Klausner “when the judge asked if string theory, Cottrell’s focus at Caltech, is “an area of physics.’

“It’s the area of physics,’ Cottrell said.”

His lawyers “attributed his odd behavior in testifying to Asperger’s syndrome”, a mild form of autism.

Funny, Cottrell isn’t the only one who goes on like this about string theory. Maybe there’s a lot of autism going around.

The latest trend among prominent theorists seems to be the writing of popular books hyping the unsuccessful speculative ideas they have been working on. Two new examples of this have been pointed out by Lubos Motl over at sci.physics.strings.

Both of these books are due to appear at the beginning of next May. One, by Leonard Susskind of Stanford, is entitled An Introduction To Black Holes, Information And The String Theory Revolution: The Holographic Universe. The second, by Lisa Randall of Harvard is called Warped Passages : Unraveling the Mysteries of the Universe’s Hidden Dimensions.

Randall’s book presumably is not so much about string theory as about the idea that we live on a brane inside a higher dimensional space. As far as I can tell, there’s even less evidence for this idea than there is for string theory itself. I don’t know exactly what her attitude about string theory is, but at a public debate at the Museum of Natural History here in New York a few years ago, I remember that she scornfully dismissed the argument that string theory predicts gravity, saying something like “Yeah, it predicts ten-dimensional gravity.”

From Sean Carroll’s weblog I see that he’s in Austin now for a session at a meeting of the Philosophy of Science Association. Philosophers of science seem to actually write up their talks in advance, and many of the talks for this meeting are already available online. Poking around on their website with papers from earlier conferences, I ran into one on Scientific Realism and String Theory by Richard Dawid.

Dawid appears to have swallowed the hype about string theory hook, line and sinker. He believes that string theory exhibits a new paradigm of how to do physics, one where the idea of being able to calculate anything about the real world and compare it to observations is passe. All that matters now is “theoretical uniqueness”, that one’s theory is the only possible one. He doesn’t seem to notice that there’s something kind of funny about people claiming that they have a wondrous unique theory, but don’t know quite what it is and can’t calculate anything about the real world with it. The S-matrix theorists of the 60s also promoted the idea that they had a wondrous unique theory, but didn’t know quite what it was. Probably one can dig up philosophy of science articles from that period about how a whole new paradigm of how to do science was required.

Dawid also seems to believe that the dualities of M-theory imply the “dissolution of ontology”, that “the ontological object has simply vanished”. In reality, what has vanished is not the ontological object, but the theory.

Over at Robert Helling’s web-site you can read an example of the latest philosophical excuses about why string theory now can’t predict anything, together with implausible wishful thinking about how this might change since “It’s just at this stage we are not yet powerful enough to make these kinds of predictions”.

For the life of me I can’t figure out why smart physicists and philosophers can’t see the obvious fact that is staring them in the face. You don’t need a new paradigm of how to do science, the old one works just fine. If you have a conjectural theoretical scientific idea, there are two ways in which it can turn out to be wrong. Either it predicts something that disagrees with experiment, or it is so vacuous that it predicts nothing. The evidence is now overwhelming that, if string theory is consistent at all, it is wrong for the second reason.

Update: Dawid actually has a whole fancy web-site about Realism and String Theory. He also has a newer paper on Undetermination and Theory Succession from a String Theoretical Perspective.

The PhilSci archive does have another paper about string theory, one by Reiner Hedrich entitled Superstring Theory and Empirical Testability. Hedrich is much less credulous than Dawid, noting about superstring theory “above all, it has fundamental problems with empirical testability – problems that make questionalbe its status as a physical theory at all.”

The KITP at Santa Barbara is holding a conference on QCD and String Theory this week, and the talks have started to appear online.

Of the ones I’ve taken a quick look at so far, there doesn’t seem to be any obvious recent progress on the 30-year old main question that everyone would like the answer to: can one find a reliable analytical technique for dealing with QCD in the infrared regiion where the effective coupling is strong? The best hope for this in recent years has been the AdS/CFT correspondence, but after seven years the state of the art there still seems to be a long ways from solving the problem one wants to solve (although it does give solutions to other problems). I’m looking forward to seeing what some of the later talks will have to say, including Larry Yaffe’s one tomorrow on “Large N gauge theories: old and new”.

Martin Veltman gave a colloquium talk at Fermilab two weeks ago and, as usual, had some very provocative comments to make. At the end of his talk he made the claim that the only thing astrophysics has contributed to particle physics is information about the number of neutrinos (from Helium abundance observations). He claims “Apart from this, Astrophysics is so far useless to us.”

He then gave some purported data about how particle physicists really felt about the impact of astrophysics and cosmology on their field. His slides say:

“Question put to many particle physicists: Do you feel that astrophysics and particle physics are joined at the hip?

Response:

Refusing to respond on the grounds that it is an obscene proposition (99.9%)
Do not know what you are talking about (9.671%)
Undecided (rest)

Questions put to particle experimenters:

Your experiment is justified by claiming that it will tell us about the first seconds of the big bang. Do you agree?

Response:

No (98.312%)
Do not know what you are talking about (1.671%)
Undecided (rest)

Do you feel that we need a new machine (linear collider) because it can be used to discover dark matter (dark energy)?

Response:

No (98.312%)
Do not know what you are talking about (1.671%)
Is this related to the death star of Darth Vader? (3%)
Undecided (rest)”

I think Veltman has a very good point. The particle physics community seems to have decided to try and sell the public on supporting particle physics, specifically a new linear collider, by claiming that such a machine will “solve the mystery of dark energy”, find “extra dimensions of space”, and tell us “how the universe came to be” (see for instance the HEPAP Quantum Universe report). This all sounds very sexy, but there’s no good reason to believe that a linear collider will do any of this. Maybe this is the right way to sell the linear collider, but personally I’m rather uncomfortable with this level of hype and wouldn’t want to be the one testifying under oath before Congress about this.

Veltman also comments that “It appears to me that the only viable solution is that this machine will be located in the US”, but given the massive deficit the Bush administration has created and current political realities, I find it hard to believe we’ll see the kind of budget increases for particle physics that would be required to make this happen anytime soon.