Not Even Wrong

The Abel prize is a new yearly prize in mathematics, intended to function somewhat like a Nobel Prize for mathematics. The first one was awarded last year to Jean-Pierre Serre and this year’s has gone to Sir Michael Atiyah and Isadore Singer, specifically for their development of the Atiyah-Singer index theorem.

Atiyah and Singer are great heroes of mine, especially Atiyah. They have both taken a great interest in the relation of mathematics and quantum field theory and are responsible for much of the fruitful exchange of ideas between the two subjects over the last 25 years. I consider much of my mathematical education to have come from a lot of time spent reading through Atiyah’s five-volume collected works. His interests have ranged over a wide swath of modern mathematics and his writing is always a model of clarity. A good case could be made that Atiyah has been the most important figure in mathematics during the second half of the twentieth century.

The Atiyah-Singer index theorem is perhaps the single most important theorem of the last half century. It links together analysis, topology, geometry and representation theory in a fundamental and surprising way, one that is dear to any physicist since it involves the Dirac operator. Very roughly, what Atiyah and Singer discovered was that the dimension of the space of solutions of certain PDEs on compact manifolds was a topological invariant, one that they could explicitly compute in terms of more well-understood topological invariants: the cohomology classes of the manifold.

They did this by noticing that the general case could be reduced to the case of the Dirac operator, twisted by the various possible vector bundles on the manifold. They actually rediscovered the Dirac operator for themselves in the course of their research. The natural abstract framework for these new topological invariants is K-theory, where classes are represented by vector bundles, much as cohomology classes are represented by differential forms.

Things get even more interesting if there is a symmetry group acting on the whole set-up. Then the space of solutions carries not just an integer, the dimension, but is a representation of the group. You can actually use the Atiyah-Singer index theorem to classify and construct geometrically the representations of many classes of Lie groups, or going the other way, use representation theory to get more powerful topological invariants. The explicit cohomological formulas you get in these cases often have versions which localize to fixed points of the group action; so you can find your answer just by locating the fixed points and looking at what happens in a neighborhood about them.

I hope Atiyah and Singer enjoy their shared $875,000!

One can keep track of what is going on in theoretical physics now by taking a look at conference websites. Often after the conference they put up speaker’s transparencies or even audio or video of the talk. Some very recent examples:

Strings and Cosmology

a conference last week at Texas A and M, and

Spring School on Superstring Theory and Related Topics

at the ICTP in Trieste. Among the Trieste lectures, Marcos Marino’s notes give a nice discussion of some things you can do with topological strings. Brandenberger’s notes on “Challenges in String Cosmology” include the peculiar statement that “String cosmology does not exist because non-perturbative string theory is not yet known”. That was my impression too, but this doesn’t really explain why he is lecturing on a subject that doesn’t exist or why people devote many conferences to it.

Among the new papers appearing at the arXiv is Witten’s latest:
Parity Invariance for Strings in Twistor Space

I’m still kind of not seeing why Witten and others are so interested in this. Using strings as a dual to QCD to understand its strong coupling behavior is obviously interesting, but why is reformulating something you understand well (perturbative Yang-Mills) in terms of strings in a super-version of twistor space so interesting? The interesting thing about twistors always seemed to me that they were naturally parity asymmetric, one chirality of spinors is tautologically defined. Witten’s latest paper seems to just be showing how to get rid of this natural chiral asymmetry in this case.

Another new paper is:

The Emergence of Anticommuting Coordinates and the Dirac-Ramond-Kostant operators

by Lars Brink. The Kostant version of the Dirac operator is pretty amazing and too little known, both among mathematicians and physicists. I’m not so sure what Brink is trying to do with it leads anywhere, but there are other applications of it I’ll try and write about some day.

I was pleased to get a comment from cosmologist Sean Carroll frighteningly soon after starting this thing up. Here’s some questions about cosmology that have been on my mind recently, maybe he or someone else will be able to answer them:

Witten has argued to me that “results about CMB fluctuations which are suggestive of inflation at the GUT scale” provide evidence that “grand unification is on the right track”. What exactly does the CMB data say about inflation? Can one extract the GUT scale from either the current CMB data or any conceivable better future CMB data?

More generally, while I’ve heard a lot about attempts to extract information about Planck-scale physics from CMB data, what about all the scales in between where accelerators stop (hundreds of Gev) and the Planck scale? Can one find out anything about electroweak symmetry breaking? Could this have anything to do with inflation?

There’s another question that often bothers me: “How can you have a field called ‘String Cosmology’ when string theory isn’t really a theory and can’t be used to predict anything?”, but I’ll be good for now and leave that rant for another time.

It’s pretty common these days for people to refer to successfully quantizing general relativity as “the Holy Grail of Physics”, but it seems to me that there is a different problem that better deserves this name:

“Why does the vacuum state break electroweak gauge symmetry?”

If we could answer this question, we’d probably understand where masses of particles come from, as well as just about all of the undetermined parameters of the standard model (except for a couple ratios of the strengths of the gauge interactions). The exciting thing about this problem is that we have good reason to expect experiments to give us some new clues about it in 2008 when data from the LHC begins to come in.

The standard unification paradigm these days explains this in terms of the potential for a Higgs field, with various grand and super-unification schemes allowing an appropriate Higgs field but somehow never being able to predict anything at all about it. Even worse, such schemes not only don’t explain anything about this field, but also require the addition of extra Higgs fields beyond the single one required by the standard model.

An idea I’ve always found appealing is that this spontaneous gauge symmetry breaking is somehow related to the other mysterious aspect of electroweak gauge symmetry: its chiral nature. SU(2) gauge fields couple only to left-handed spinors, not right-handed ones. In the standard view of the symmetries of nature, this is very weird. The SU(2) gauge symmetry is supposed to be a purely internal symmetry, having nothing to do with space-time symmetries, but left and right-handed spinors are distinguished purely by their behavior under a space-time symmetry, Lorentz symmetry. So SU(2) gauge symmetry is not only spontaneously broken, but also somehow knows about the subtle spin geometry of space-time. Surely there’s a connection here…

This idea has motivated various people, including Roman Jackiw, who has several papers about chiral gauge theories that are very much worth reading. The problem you quickly get into is that the gauge symmetry of chiral gauge theories is generally anomalous. People mostly believe that theories with an anomalous gauge symmetry make no sense, but it is perhaps more accurate to say that no one has yet found a unitary, Lorentz-invariant, renormalizable way of quantizing them. In the standard model, the contributions to the anomaly from different particles cancel, so you can at least make sense of the standard perturbation expansion. Outside of perturbation theory, chiral gauge theories remain quite mysterious, even when the overall anomaly cancels.

So, this is my candidate for the Holy Grail of Physics, together with a guess as to which direction to go looking for it. There is even a possible connection to the other Holy Grail, I’ll probably get around to writing about that some other time.

Last Friday I went to hear a talk by David Gross at the CUNY Graduate Center on “The Coming Revolutions in Fundamental Physics”. This was more or less Gross’s standard advertisement for string theory that he has been giving for nearly 20 years now. He explicitly started off with the claim:

“Fundamental Physics = String Theory”

His first PowerPoint slide was a quote from “Hannibal” by Thomas Harris, a sequel to the novel “Silence of the Lambs”. Evidently late in this novel the main character is depicted manipulating the “symbols of string theory”, with “equations that begin brilliantly and end in wishful thinking”. It sounds like Harris really has got the right idea about string theory, but Gross said his talk was designed to argue that string theory was not wishful thinking.

He then went on to claim that in 3-4 years there will be a headline in the New York Times about the discovery of supersymmetry at the LHC. From what I can tell, the LHC should have first beams in 2007, but even if everything goes according to plan, it won’t be until 2008 that the experiments there will have accumulated a non-trivial amount of data and analyzed it. Experience with colliders generally has been that getting them running at a useful luminosity can take quite awhile after they are first turned on. So 3 years from now is definitely out, 4 years is optimistic. This is now getting close enough that Gross and others seem intent on ignoring the failures of string theory, desperately hoping that superpartners will pop out of the LHC, thereby providing at least some vindication of the train of reasoning that lead to string theory. What will be interesting to see will be what Gross et. al. do when this doesn’t happen. Will they drop string theory? Quite possibly the LHC will revolutionize physics by showing us what is really causing the spontaneous breaking of the electroweak gauge symmetry. If this happens, everyone will abandon string theory and start working on this, 1984-2008 then becoming a period in the history of physics that particle theorists try and not think about.

Gross’s talk contained the usual tendentious pro-string theory points, here’s a few of them with commentary:

1. ” String theory is in a period like that of 1913-1925, it’s like the Bohr model, we’re waiting for the analog of Heisenberg’s or Schrodinger’s breakthroughs”

The problem with this is that the Bohr model was actually predictive, for instance it predicted a lot about atomic spectra that could be experimentally checked. There clearly was something right about the Bohr model, there is no good evidence there is something right about string theory.

2. “String theory is better than QFT, because QFT Feynman diagrams have these interaction vertices you can assign any interaction strength you want to”

This is not true of gauge theories, the different vertices are related by gauge symmetry. True you have to pick over-all gauge groups and representations, and the Higgs sector is problematic, but the claim that there is just one string theory is just wishful thinking.

3. “String theory is better than QFT because interactions are not at points, so short distance behavior is better”

Gross should be well aware that asymptotically free gauge theories are extremely well-behaved at short distances despite having point-like interactions, since he discovered this. It is also true that string theory perturbation theory is only known to be well-behaved up to two loops. My colleague Phong claims that higher loops remain very much not understood.

4. “String theory is a consistent, finite quantum theory of gravity”

Simply not true. Peturbative string theory is a divergent expansion, non-perturbative definitions don’t work for four large flat dimensions, rest small.

5. “String theory inspired brane-world scenarios, although I don’t really believe these”

Why would you think that an argument in a theory’s favor was that it inspired some clearly wrong models that you don’t believe and that don’t predict anything?

While Gross mentioned the “discretium”, he didn’t really explain exactly how disastrous this is for string theory, since it makes it essentiallly vacuous. He made a big deal of string theory implying that our notions of space and time need to be changed, but made it clear that no one really has a viable idea about how to change them. He puts his hopes in the fact that we still don’t understand what string theory is. This seems to me to be exactly the sort of wishful thinking that he claimed at the beginning was not what string theorists were doing.

His talk went on for more than an hour and a half. Several questions from the audience were taken, including one from Michio Kaku who claimed that dark energy was evidence of supersymmetry and asked about theories with two times. Gross didn’t seem interested in saying much about such theories. I noticed that two string theory postdocs I know were in the audience. They’ve both told me that they think the subject is at a point of crisis and they are thinking of quitting. I don’t think anything Gross said was likely to encourage them to continue.

This is an experimental new weblog. Let’s see if I can find anything of interest to put on here, and how long it takes for someone else to find out it exists.