Not Even Wrong

Sean Carroll has some comments about the anti-big bang petition. He also points to Ned Wright’s explanation of what is wrong with various proposed alternatives to the big bang scenario. In particular this explains in detail what the problems with Irving Segal’s “Chronometric Cosmology” are, something I’d always wondered about.

Segal was a very good mathematician, who did serious work on quantum field theory in the 50s and 60s. He’s the “Segal” in what is sometimes called the “Segal-Shale-Weil” representation. Segal is a counter-example to Carroll’s observation that, for the most part, opponents of the big bang are not very smart. Unfortunately, it seems that quite smart and otherwise reasonable people can have unshakable faith in ideas that don’t work. Segal’s student John Baez wrote up some of his memories of his advisor and his cosmological research.

Slides used by many of the lecturers at the recent Davis mathematical physics conference in honor of Albert Schwarz are now online.

It’s always a little worrying when this happens, but sometimes I find myself very much agreeing with at least parts of what Lubos Motl has to say. For example see this recent posting to sci.physics.strings. In it Motl argues that

“I think it is premature to try to construct this major framework that would explain the character of vacuum selection and very early cosmology in string theory”

and

“So my belief is that we will have to understand the nonperturbative structure of the stringy arena using some new universal definition of string theory – a definition that is both non-perturbative (reaches the strongly coupled regions) as well as background-universal (reaches the geometries and non-geometries that are different from the starting one), and only afterwards, we will be able to start answering the stringy cosmological questions in a better context. Without this new tool, everything is just vague guesswork.

In my opinion, the research of string cosmology; stringy inflation; de Sitter space in string theory; scattering in backgrounds with non-standard causal diagrams; and all similar things that have essentially be started by the observation of accelerating Universe back in 1998 – has led to a very small number of intriguing results. There is almost nothing non-trivial and mathematically intriguing going on here; there is as much output as much input we insert. It remains a combination of phenomenology and speculations where conjectures can rarely be clearly ruled out.”

I’ve never really understood why there are fields of “string phenomenology” and “string cosmology” when the theory is still in a state that it can’t reliably calculate anything. While I think it is wishful thinking to believe that if you understood string theory better it would reproduce the real world, at least Motl’s is a consistent scientific position.

Motl is also a fierce opponent of the “anthropic” arguments that have become popular among string theorists. For the latest example of anthropic argumentation, see this posting at Jacques Distler’s weblog.

For many years now the SPIRES database at SLAC has been used to produce a list of the most frequently cited papers during each year. Since 1997 Michael Peskin has been doing this, while at the same time writing up a description of what is in the 40 or so most popular papers, together with comments on what this data shows about trends in particle physics. The 2003 edition of Peskin’s review has recently appeared.

Peskin notes that SPIRES has begun indexing more astrophysical papers during the last two years, and many particle physicists have turned their attention to cosmology. He has expanded the number of top papers he reviews from 40 to 50 to take into account the greater coverage of the database.

The most frequently cited article, this year and every year, is the Particle Data Group’s “Review of Particle Physics” compilation of experimental particle physics data. It is conventional for experimental papers to often refer to this instead of to the original papers. This year the number two and three positions are held by papers from the WMAP experiment, with number four the original results on high redshift supernovae that indicated a non-zero cosmological constant.

The first particle theory paper is the Randall-Sundrum one at number five, and Maldacena’s AdS/CFT paper is at number seven. For many years the top part of this list was heavily dominated by relatively new string theory papers, but the situation is now dramatically different. The highest-ranked post-1999 paper is one about PP-waves at number 18, the next is one at number 37 by Ashoke Sen about time-dependent backgrounds. The only other post-1999 paper in the top 50 is the Dijkgraaf-Vafa paper about supersymmetric gauge theories, which is at number 39.

This list provides pretty conclusive evidence that the field of particle theory more or less flat-lined about 5 years ago, with only a small number of minor blips of brain activity since then.

There’s also a cumulative list of the most highly cited papers of all time. Here the dramatic movement one can watch is the speed with which Maldacena’s paper accumulates citations. At the end of last year it was at number 6 on the list of all-time most frequently cited papers; it has now moved to number 5 and soon will overtake number 4. Within a couple of years it should be at number three, only outranked by the Review of Particle Properties and Weinberg’s original paper on the Weinberg-Salam model.

The hundreds of expository articles about supersymmetry written over the last twenty years or more tend to begin by giving one of two arguments to motivate the idea of supersymmetry in particle physics. The first of these goes something like “supersymmetry unifies bosons and fermions, isn’t that great?” This argument doesn’t really make a whole lot of sense since none of the observed bosons or fermions can be related to each other by supersymmetry (basically because there are no observed boson-fermion pairs with the same internal quantum numbers). So supersymmetry relates observed bosons and fermions to unobserved, conjectural fermions and bosons for which there is no experimental evidence.

Smarter people avoid this first argument since it is clearly kind of silly, and use a second one: the “fine-tuning” argument first forcefully put forward by Witten in lectures about supersymmetry at Erice in 1981. This argument says that in a grand unified theory extension of the standard model, there is no symmetry that can explain why the Higgs mass (or electroweak symmetry breaking scale) is so much smaller than the grand unification scale. The fact that the ratio of these two scales is so small is “unnatural” in a technical sense, and its small size must be “fine-tuned” into the theory.

This argument has had a huge impact over the last twenty years or so. Most surveys of supersymmetry begin with it and it justifies the belief that supersymmetric particles with masses accessible at the LHC must exist. Much of the experimental program at the Tevatron and the LHC revolve around looking for such particles. If you believe the fine-tuning argument, the energy scale of supersymmetry breaking can’t be too much larger than the electroweak symmetry breaking scale, i.e. it should be in the range of 100s of Gev- 1 Tev or so. Experiments at LEP and the Tevatron have ruled out much of the energy range in which one expects to see something and the fine-tuning argument is already at the point of starting to be in conflict with experiment, for more about this, see a recent posting by Jacques Distler.

Last week at Davis I was suprised to hear Lenny Susskind attacking the fine-tuning argument, claiming that the distribution of possible supersymmetry breaking scales in the landscape was probably pretty uniform, so there was no reason to expect it to be small. He believes that the anthropic explanation of the cosmological constant shows that the “naturalness” paradigm that particle theorists have been invoking is misguided, so there is no valid argument for the supersymmetry breaking scale to be low.

I had thought this point of view was just Susskind being provocative, but today a new preprint appeared by Nima Arkani-Hamed and Savas Dimopoulos entitled “Supersymmetric Unification Without Low Energy Supersymmetry and Signatures for Fine-Tuning at the LHC“. In this article the authors go over all the problems with the standard picture of supersymmetry and describe the last twenty-five years or so of attempts to address them as “epicyclic model-building”. They claim that all these problems can be solved by adopting the anthopic principle (which they rename the “structure” or “galactic” or “atomic” principle to try and throw off those who think the “anthropic” principle is not science) to explain the electroweak breaking scale, and assuming the supersymmetry breaking scale is very large.

It’s not suprising you can solve all the well-known problems of supersymmetric extensions of the standard model by claiming that all effects of supersymmetry only occur at unobservably large energy scales, so all we ever will see is the non-supersymmetric standard model. By itself this idea is as silly as it sounds, but they do have one twist on it. They claim that even if the supersymmetry breaking scale is very high, one can find models where chiral symmetries keep the masses of the fermionic superpartners small, perhaps at observably low energies. They also claim that in this case the standard calculation of running coupling constants still more or less works.

The main experimental argument for supersymmetry has always been that the running of the three gauge coupling constants is such that they meet more or less at a point corresponding to a unification energy not too much below the Planck scale, in a way that works much better with than without supersymmetry. It turns out that this calculation works very well at one-loop, but is a lot less impressive when you go to two-loops. Read as a prediction of the strong coupling constant in terms of the two others, it comes out 10-15% different than the observed value.

I don’t think the argument for the light fermionic superpartners is particularly compelling and the bottom line here is that two of the most prominent particle theorists around have abandoned the main argument for supersymmetry. Without the pillar of this argument, the case for supersymmetry is exceedingly weak and my guess is that the whole idea of the supersymmetric extension of the standard model is now on its way out.

One other thing of note: in the abstract the authors refer to “Weinberg’s successful prediction of the cosmological constant”. The standard definition of what a prediction of a physical theory is has now been redefined down to include “predictions” one makes by announcing that one has no idea what is causing the phenomenon under study.

Slate this morning has an article by Jim Holt about an interview with Andrei Linde. In the interview, Linde speculates that universes like ours could be created in a lab, that maybe we live in such a universe, and that the creator of such a universe could communicate with his/her creations by tuning the parameters of its “Landscape”.

The last two talks at the Davis conference were quite interesting. Alexandre Givental gave one entitled “Twisted Loop Groups and Gromov-Witten Theory”, which went by way too fast. He has an interpretation of the generating functional for Gromov-Witten invariants that uses a loop group. Some infinite dimensional symplectic geometry involving this group gives a conjectural explanation of the properties of these invariants. The talk covered a lot of material so, while intriguing, was quite hard to follow.

Witten gave the last talk, and his philosophy was the opposite of Givental’s. He covered only a little material, at a level that was easy to follow. The nominal topic of his talk was his recent work on the relation of strings in twistor space to gauge theory scattering amplitudes, but he didn’t really get to this. He began by saying that there were at least a couple different possible talks he could give about the background to his recent work. One was the one he gave a couple weeks ago at a conference at NYU which covered gauge theory scattering amplitudes. Luckily for me since I had heard that one, he decided to give a different one at Davis, mostly covering some of the ideas about twistors used in his work. He discussed the twistor theory construction of an SU(2,2) representation using the massless single particle solutions of a fixed helicity. This is related to a non-compact version of the Borel-Weil-Bott theorem, constructing a representation on a higher cohomology space. Presumably one could also do this using the Kostant Dirac operator techniques I mentioned recently.

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.