Not Even Wrong

I’m in Nothern California this week, and have been attending some of the talks at the conference at UC Davis celebrating Albert Schwarz’s 70th birthday. The landscape at Davis is exceedingly flat, but this morning Lenny Susskind gave a remarkable talk with the title “Exploring the Landscape”.

It was a pretty strange talk for a mathematical physics conference since it contained zero mathematics (and it’s arguable whether there was any physics…). Susskind blamed Iz Singer for this, claiming that Singer told him he should talk about the landscape stuff since it was leading to a new mathematical field of “statistical topology”. He began by holding up a copy of Steven Weinberg’s “Dreams of a Final Theory” and reading a quote from it about the cosmological constant. He liked this so much he read the same quote a second time a little while later.

He then discussed some of the recent history of string theory, noting that for a long time string theorists were hoping for a mathematical silver bullet that would provide a more or less unique solution to the theory that looked like the real world. He announced that now the probability of this is less than 1 in 10^500.

Susskind then explained a bit about KKLT vacua, saying that his main reason for discussing them was to show how silly and inelegant they are. He compared them to a Rube Goldberg machine and called Shamit Kachru the “master Rube Goldberg architect”.

The most dramatic part of Susskind’s talk was something new: an attack on the idea of low-energy supersymmetry. He explained the standard fine-tuning argument for supersymmetry, but then indicated that he thought an anthropic argument made more sense. The reason the Higgs mass is so much smaller than the Planck mass is not supersymmetry, but instead because that small size is necessary for our existence. He said that the question of low-energy supersymmetry is something that Douglas’s statistical analysis of vacua should address (Douglas will talk tomorrow), but his view is that low-energy supersymmetry will be very unlikely.

In the question session, John Schwarz challenged him about this, claiming that there were other reasons to believe in low-energy supersymmetry, including the unification of coupling constants and the idea that dark matter is the lowest mass superpartner. Susskind’s response was that even though there were a couple reasons like those, there were many more that indicated problems with the idea of low-energy supersymmetry, including problems with too fast proton decay.

It was pretty amazing to see someone challenging the supersymmetry orthodoxy. On the other hand, the whole program Susskind and others are pursuing is completely loony. String theory predicts absolutely nothing, and instead of drawing the obvious conclusion that it is a useless idea, Susskind is trying to turn this failure into some perverse sort of virtue.

Update: In Michael Douglas’s talk today he said that his calculations show no reason for a low-energy supersymmetry breaking scale to be especially likely. So he expects that supersymmetry will only be broken at a high energy. Maybe somebody should tell the people working on the LHC experiments that the whole supersymmetry thing is now off, they should find something else to look for.

Quantum mechanics and representation theory are very closely linked subjects since the Hilbert space of a quantum system with symmetry group G carries a unitary representation of G. To the extent that one has a way of quantizing a classical Hamiltonian system with G-symmetry, one has a way of constructing representations of G out of symplectic manifolds with G-action. This “geometric quantization” approach to constructing representations has been a very fruitful one.

For the case of G compact, connected, with maximal torus T (the crucial example to keep in mind is G=SU(2), T=U(1)), the “flag manifold” G/T (the 2-sphere for G=SU(2)) is a symplectic manifold (actually Kahler) and can be thought of as a classical phase space with G-symmetry. Choosing a representation of T (a “weight”) allows one to construct a line bundle over G/T, which turns out to be holomorphic. The Borel-Weil theorem says that irreducible G-representations are given by holomorphic sections of this line bundle, for “dominant” weights.

For weights that are not dominant, one gets not holomorphic sections, but elements in higher cohomology groups. These can be expressed either in terms of the sheaf cohomology of G/T with coefficients in the sheaf of holomorphic sections of the line bundle, or in terms of Lie algebra cohomology. This is known as the Borel-Weil-Bott theorem, which first appeared in:

Bott, R., Homogeneous Vector Bundles, Ann. of Math. 66 (1967) 203.

the Lie algebra version was further developed by Kostant in

Kostant, B., Lie Algebra Cohomology and the Generalized Borel-Weil Theorem, Ann. of Math. 74 (1961) 329.

Instead of using complex manifold methods and the Dolbeaut operator to construct cohomology classes, one can use spinors and the Dirac operator, with the representation appearing as the kernel of the Dirac operator (or, more accurately, its index). For this point of view, which fits in beautifully with equivariant K-theory and the index theorem, see:

Bott, R., The Index Theorem for Homogeneous Differential Operators, in: Differential and Combinatorial Topology: A Symposium in honor of Marston Morse, Princeton (1964) 71.

The Dirac operator approach to representation theory has been extended to some cases of G non-compact by various authors. In the last few years, Kostant has come up with a new version of the Dirac operator in this context which has quite interesting properties. He likes to work algebraically, so his Dirac operator on G is given as an element of U(Lie G)XCliff(Lie G), where U(g) is the universal enveloping algebra of the Lie algebra Lie G and Cliff(Lie G) is the Clifford algebra of Lie G. The Kostant Dirac operator is the standard one you would expect, with the addition of an extra cubic term. For the details of all this, see Kostant’s paper:

Kostant, B. , A Cubic Dirac Operator and the Emergence of Euler Number Multiplets of Representations for Equal Rank Subgroups, Duke Math. J. 100 (1999) 447.

Things get interesting when you consider the case of H a subgroup of G of the same rank (one example is H=T, another important one is G=S0(2n+1), H=SO(2n), where G/H is an even-dimensional sphere). Taking the difference of Kostant Dirac operators for G and H gives something that corresponds to a Dirac operator on G/H, which acts on the product of a G rep with the spinors associated to Cliff (Lie G/Lie H). For H=T, one gets back the old Bott-Kostant construction of representations, but with the Lie algebra cohomology replace by the index of a Dirac operator.

Part of this story is that one finds that, starting with an irreducible G-representation, the kernel of Kostant’s Dirac operator consists of a “multiplet” of H representations of size given by the Euler characteristic of G/H. The existence of these multipliets was first noticed by Ramond for the case H=SO(9), where SO(9) is the massless little group in 11 dimensions and the multiplets appear in the massless spectrum of N=1 11d-supergravity (the low energy limit of a conjectural M-theory). These SO(9) multiplets come about because SO(9) is an equal rank subgroup of the exceptional group F4, so for each irreducible F4 representation one gets a multiplet of SO(9) representations.

The first paper about this was by Gross, Kostant, Ramond and Sternberg, for more about this from a geometrical point of view, see a paper by Greg Landweber. For a discussion of the relation of this to supersymmetric models in physics, look up recent preprints by Pierre Ramond, one of which is by Brink and Ramond.

Greg Landweber has applied these ideas to loop groups, getting a beautiful interpretation in terms of loop group representation theory of certain N=2 superconformal models first studied by Kazama and Suzuki in 1989. This paper also contains a detailed exposition of the story both for finite dimensional groups and loop groups.

More recently, Freed, Hopkins and Teleman have used a modified version of the Kostant Dirac operator to give a proof of their theorem relating the Verlinde algebra and twisted K-theory. Their construction is quite beautiful and gives a new point of view on the whole story of the relation of geometric methods of quantization to K-theory and the index of Dirac operators. I’ll try and write something about this at some later date.

For quite a few years now, when I ask my colleague Herve Jacquet about what is going on in his field, he tells me something like: “Maybe someone will soon be able to prove the Fundamental Lemma”. This is a bit of a joke since the terminology “lemma” is supposed to refer to an easy to prove, simple technical result needed on the way to proving a real theorem. In this case the name “Fundamental Lemma” has ended up getting attached to a crucial conjecture that is part of the so-called “Langlands Program”, and this conjecture has resisted all attempts to prove it for more than twenty years.

A few weeks ago Jacquet told me that he had heard that two French mathematicians, Gerard Laumon and Bao Chau Ngo, finally had a proof and today a manuscript has appeared on the arXiv. The techniques it uses are way beyond me (and even Jacquet claims he doesn’t understand them), but are related to those used in the so-called “Geometric Langlands Program” that has interesting relations to conformal field theory.

I won’t embarrass myself by trying to explain in any detail the little that I know about this kind of mathematics, but in extremely vague terms the Langlands Program relates representations of the Galois group (which tell one about the number of solutions of arithmetic problems) to representations of algebraic groups like the general linear group. One example of this kind of thing is the Taniyama-Shimura-Weil conjecture that was proved by Wiles and implies Fermat’s Last Theorem. One way of approaching the Langlands program uses generalizations of the Selberg trace formula, and the lack of a proof of the fundamental lemma has evidently been the main obstruction to getting all that one would like out of the trace formula methods. Maybe someday I’ll understand some of this enough to try and write something more, but that will probably take quite a while. In the meantime, one of the few expository papers I’ve found about the fundamental lemma is here.

Alvaro de Rujula has posted on the arXiv under the title “Fifty years of Yang-Mills Theories: a phenomenological point of view” some of his recollections from the mid-seventies. These bring back my own memories of taking a course on particle theory from him at Harvard around 1977-78. One amusing aspect of the course was that when introducing a concept carrying someone’s name, de Rujula would always say something like “this is the so-called Weinberg angle, which of course was discovered by Glashow”. In one lecture he did something a bit different, saying something like “this is the Cabibbo angle, which, strangely enough, I think actually may have been discovered by Cabibbo”. de Rujula’s paper contains one of his famous drawings from the period and an amusing picture of Georgi and Glashow arguing. His asides are entertaining, but some so obscure I confess to not knowing exactly what he is referring to.

Today’s arXiv postings also contain a review talk on the state of string theory. It discusses the “landscape” with the comment “However, with such large numbers of vacua involved, one must wonder whether the scheme is at all testable, even in principle.” Normally string theory reviews start by describing the theory as the “only known” or “best candidate” or “most promising” approach to unification. This one replaces those phrases by “dominant framework”, and one certainly can’t argue with that.

The Theoretical Particle Physics Jobs Rumor Mill has a new home. It’s no longer at the University of Washington, now it’s at the College of William and Mary Physics department.

Now that it’s available again, the Rumor Mill has the striking news that Harvard has chosen for a faculty position one of its postdocs: Lubos Motl. Lubos is well-known as undoubtedly the most rabidly fanatic string theorist around, always willing to heap abuse and scorn on anyone who questions the idea that string theory is the language in which God wrote the world. Unlike many string theorists though, he actually knows what is going on in the field and is someone who can give you an accurate view of exactly what the state of the theory is (all you have to do is strip out his ravings about how string theory is unique and the source of all good ideas in physics and mathematics). He’s also not foolish enough to swallow the “Anthropic” nonsense that is becoming ever more prevalent among string theorists, and it’s a been a bit scary recently to see him acting as the voice of reason in the subject.

HEPAP is the Department of Energy’s “High Energy Physics Advisory Panel”, which holds meetings 3-4 times a year. At these meetings, people from the DOE and NSF report on the latest news about US government funding for particle physcs, and physicists from the universities and national labs report on how their experiments are going.

The latest HEPAP meeting was held this past weekend in Washington, and some of the presentations there have already been made available online. These include a detailed report on the progress of the LHC and the two experiments (CMS and Atlas) that will do physics there. The LHC construction is 90% complete, with magnets beginning to be installed in the tunnel. Things seem to be on track for turning on the machine in the spring of 2007. Optimistically, this would mean the first physics results should be available sometime in 2008.

Another presentation gave an overview of the DOE’s support of university-based particle physics. This includes the largest source of support in the US for theoretical particle physics. In FY 2003 the DOE spent \$23.3 million supporting theory research at 68 universities, funding 215 faculty, 116 postdocs and 114 graduate students. The other main source of US funding for particle theory is the NSF, which spends about half as much as the DOE (\$12 million in FY 2003).

HEPAP also adopted a new report on the “Quantum Universe“. This follows the trend of recent years of trying to justify particle physics research by emphasizing its relation to the very healthy and sexy field of cosmology. The acid test of this over the next few years will be to see if it helps with the difficult problem of getting funding for the Linear Collider.

It seems that the second most important web-site in the particle theory community (the first is obviously the arXiv ) has been shut down by the University of Washington. The Theoretical Particle Physics Jobs Rumor Mill has for years been a comprehensive source of information about who’s hot, who’s not, the hiring plans of all the theoretical particle physics groups in the United States and Canada, and the career moves of more established theorists. I hope a new home for the site can be found at a less fearful institution. I’d consider putting it up here, but I’ve got enough particle theorists annoyed at me already…

A more and more common argument one hears from string theorists these days (for one version see a recent anonymous comment posted here) goes more or less like this:

“A fundamental theory shouldn’t be expected to predict things like fermion masses or the standard model gauge group anymore than it should be able to predict the physical properties of the planets. Anyone who expects this is making the same mistake as Kepler, who tried to relate Platonic solids to planet orbits.”

The idea here is that many or even all of the things we don’t understand about the standard model are not fundamental aspects of the theory we should expect to be able to predict. Perhaps they are determined by the details of the history of how we ended up in this particular time and place, just as the properties of the planets were determined by the detailed history of the formation of the solar system.

As far as we can tell, the properties of the standard model hold uniformly throughout the observable universe, so to adopt this point of view one needs to postulate the existence of an unobservable “multiverse” of which we see only one small part. The so-called “landscape” of an unimaginably large number of possible vacuum solutions for string theory provides one realization of such a multiverse.

What are the problems with this idea? First of all, it is not so easy to dismiss out of hand. One can certainly imagine the possibility of the existence of an M-theory (maybe now the “M” is for “Multiverse”) with a local vacuum state that corresponds to our universe, and some dynamics that allows evolution from one universe to another. Perhaps tomorrow night a preprint will appear on arxiv.org containing a simple equation expressing a dynamics such that the possibility of a universe exactly like ours does arise as some part of a solution. Should we believe in such a new theory, whatever it is?

There seem to me to be two possible cases in which such a theory would be compelling. The first would be if the theory made experimentally testable predictions. Perhaps it would have only one solution that agreed completely with current experimental observations. Then the properties of this solution could be used to predict the results of experiments not yet done. If these predictions were accurate, the theory would have strong evidence in its favor.

Even if the theory had so many solutions that one couldn’t readily use it to make predictions, one still might find it compelling due to its “beauty” or “elegance”. If it were based on a very simple equation or idea, the fact that the relatively complex structure of the standard model could be made to fall out of a much simpler equation would again be strong evidence for such a theory. Just how compelling this would be would depend on how much simpler it was than the standard model. If the new equation was more or less as complicated as the equations which determine the standard model, it wouldn’t be compelling at all.

The current state of affairs in particle theory is that many people believe that they are on the road to finding such a compelling theory, but all the evidence is that this is nothing but wishful thinking on their part. There is no viable proposal for an M-theory based on a simple set of equations with a solution corresponding to the real world. This simply does not exist. An easy way to embarass a string theorist who is going on about the beauty of the theory is to ask them to write down a simple set of equations that characterize this beautiful theory. They can’t do it now and I don’t see any reason to believe they ever will be able to in the future.

What string theorists have now is not a single, consistent theory, but a set of several inconsistent fragmentary theories that they hope can be turned into a consistent whole. This circle of ideas is significantly more complicated than the standard model that it is trying to explain.

Even this complex of ideas might be compelling if it could be used to explain one or more not yet understood aspects of the standard model, or if it made new experimental predictions that could be checked. All the evidence of recent years is that this is impossible. If the whole framework makes any sense at all, it appears to predict nothing and explain nothing about the standard model. Not a single one of the parameters of the standard model can be calculated, not a single experimental prediction, at any energy scale, can be made. It is becoming increasingly clear that the circle of ideas known as “M-theory” is completely vacuous.

Strong evidence that this is the case comes from the fact that string theorists have no idea what, if anything, M-theory is supposed to be able to predict. Polchinski and others feel they have demonstrated that M-theory can’t predict the cosmological constant, but can’t come up with anything else it can predict and, increasingly, seem happy to live with the idea of promoting a theory that can’t predict anything. This wholesale abandonment of the scientific method upsets some physicists such as David Gross quite a bit, but more and more people seem to have no problem with this. Frankly I find this all bizarre, disturbing, and becoming ever more so all the time.