Not Even Wrong

Every year around this time the Edge web-site posts responses from a large number of scientists to some particular question. This year the question is “What have you changed your mind about? Why?”, and the results should be posted tomorrow. John Baez has posted his answer here. He writes that he changed his mind about the question of whether he should be thinking about quantum gravity after he realized “the more work we did, the more I realized I didn’t know what questions we should be asking!”, and compares the effort to “throwing darts in a darkened room and hoping to hit the bull’s-eye.” Since changing his mind he has been working on other things, and feels that as a result “I’m making more real progress understanding the universe than I ever did before.”

I think I share John’s point of view on this in many ways, even though I’ve never actively worked on a quantum gravity research project. The problem with quantum gravity research has always seemed to me that, since you can’t measure quantum gravitational effects, you’re in great danger of coming up with lots and lots of “quantum gravities”, but unable to ever know which if any of them has anything to do with the real world. This is kind of what has happened with string theory in recent years. One hope has always been that one will find a mathematically uniquely compelling model, but that has yet to happen. To me the best bet has always been that one might understand quantum gravity by unifying it with the standard model, in a compelling way such that the unified theory explains some of the things the standard model leaves unresolved. This hope was also behind much of the original interest in string theory: it wasn’t just a quantum theory of gravity, but also a theory of particle physics that could be tested.

Like John, I think we still have a long ways to go towards understanding at a deep enough level how quantum field theories really work, and how the internal symmetries of particle physics and the space-time symmetries of gravity can be unified into a more fundamental structure. Progress towards this goal may even require new mathematics, and this means there are all sorts of things to think about and work on.

One person who hasn’t changed his mind about some things is Lubos Motl. According to his latest posting, I’m

…a typical incompetent, power-thirsty, active moron of the kind who often destroy whole countries if they get a chance to do it.

The arXiv puts out charts each year showing the number of submissions by category, the ones for 2007 are now available here. Commentary about this here and here. The general trend is that quite a few years ago the number of HEP papers leveled off, as just about all of them were posted on the arXiv. The number of math papers is still growing quickly, it is only recently that posting math preprints to the arXiv has become a widespread practice. While mathematicians probably write papers at a slower rate than physicists, there are a lot more mathematicians than particle physicists.

The end of 2007 has brought one undesirable change. Recently I was down in Princeton, and found out that the University Store, traditionally one of the best places in the world to buy math and physics books, has now gone out of the book-selling business, turning it over the Labyrinth Books. The new Labyrinth store has some math and physics books, but far, far fewer. Maybe they just are getting started, and 2008 will bring better news about this.

This past summer Terry Tomboulis posted a preprint claiming to have a proof of confinement. Recently there has been a note posted on the arXiv by Ito and Seiler claiming to have found a problem with his proof, and a response from him claiming there is no problem. I’d love to hear from an expert who has taken the time to follow these arguments carefully and can explain what is going on here.

For a recent article by Arthur Jaffe surveying the history of rigorous studies of quantum field theory, see here.

Heading off late tomorrow for a 9 day vacation in Paris. Blogging will probably be light to non-existent and it will take me longer to get around to deleting comments. Please don’t feed the trolls.

Best wishes to all for the new year….

Theoretical physics and mathematics are much cheaper to fund than experimental particle physics. I don’t know yet what the implications of the FY 2008 budget are for these fields, but it appears that they are not facing large cuts like Fermilab. In other funding-related news:

DARPA, a Defense Department division responsible for funding research that might lead to new technology with military applications, is soliciting applications for grants to fund pure mathematical research. In the past they have funded work on the geometric Langlands program, now they have put out a list of 23 Mathematical Challenges, illustrating what they would like to fund. These include some conventional Clay Millennium Prize problems like the Riemann Hypothesis and Hodge Conjecture, and some less conventional ones like

Biological Quantum Field Theory
Quantum and statistical methods have had great success modeling virus evolution. Can such techniques be used to model more complex systems such as bacteria? Can these techniques be used to control pathogen evolution?

There’s even one I find extremely tempting:

Geometric Langlands and Quantum Physics
How does the Langlands program, which originated in number theory and representation theory, explain the fundamental symmetries of physics? And vice versa?

Benjamin Mann, the program officer, explains the rationale here.

On the question of grants, there are some interesting comments by Tom Banks in the comment section of this posting at Cosmic Variance. The posting quotes Harvard president Faust as warning lesser schools that they should get out of the business of scientific research, since Harvard is going to be vigorously and successfully competing for increasingly scarce government funding for such research. Banks describes how when he and others were trying to build up the string theory group at Rutgers

…we never got the kind of government funding that the elite institutions have (I’m counting dollars per person). This was at a time when certain elite institutions were at a low ebb and were getting scandalously large amounts per person in return for mediocre research. And of course, in the end, two of our most successful researchers got stolen away by elite institutions.

I guess he’s referring to Seiberg, who went to the IAS, and Shenker, who went to Stanford. I don’t know which are the “certain elite institutions” that were doing “mediocre research” that he is referring to.

The Rutgers string theory group was originally built up when the university spent large amounts of money to bring in several prominent string theorists. One version of this story that I heard was that a Rutgers official called up Dan Friedan one day in his office, and asked him what it would take to get him and several other well-known string theorists to move there. Friedan had no interest in going to Rutgers, so made up what he considered an absurd list of demands (huge salaries, lots of postdocs, new building, little or no teaching, etc., etc…). The official thanked him and then hung up, with Friedan convinced he’d never hear any more about this. A couple hours later though, the phone rang again. It was the same official, telling Friedan that they would be more than happy to meet all his demands.

Another institution that I hear is trying to compete with the elite by starting up a new, well-funded institute for research in math and physics is Stony Brook. The money is coming from Jim Simons, as part of a donation announced last year. Some other information about donations from Simons to other institutions is available at the Simons Foundation website. This foundation is largely devoted to funding research on autism, but also describes donations to the Math for America program to recruit math teachers, as well as to Brookhaven, Stony Brook, the IAS, IHES and MSRI. Simons has been funding summer workshops at Stony Brook for several years that are largely devoted to string theory. The math department has an NSF-funded RTG program in geometry and physics, and the web-site there includes links to write-ups of some of the expository talks that are part of the program.

David Vogan was visiting Columbia last week, giving the Ritt Lectures, on the topic of Geometry and Representations of Reductive Groups. He has made available the slides from his lectures here.

Vogan’s talks concentrated on describing the so-called “orbit method” or “orbit philosophy”, which posits a bijection for Lie groups G between

and

This is described as a “method” or “philosophy” rather than a theorem because it doesn’t always work, and remains poorly understood in some cases, while at the same time having shown itself to be a powerful source of inspiration in representation theory.

It is probably best understood as an expression of the deep relationship between quantum mechanics and representation theory, and the surprising power of the notion of “quantization” of a classical mechanical system. In the Hamiltonian formalism, a classical mechanical system with symmetry group G corresponds to what a mathematician would call a symplectic manifold with an action of the group G preserving the symplectic structure. “Geometric quantization” is supposed to associate in some natural way a quantum mechanical system with symmetry group G to this symplectic manifold with G-action, with the Hilbert space of the quantum system providing a unitary representation of the group G. The representation is expected to be irreducible just when the group G acts transitively on the symplectic manifold. One can show that symplectic manifolds with transitive G action correspond to orbits of G on (Lie G)^*, the dual space to the Lie algebra of G, with G acting by the dual of the adjoint action. So it is these “co-adjoint orbits” that provide geometrical versions of classical mechanical systems with G symmetry, and the orbit philosophy says that we should be able to quantize them to get irreducible unitary representations, and any irreducible unitary representation should come from this construction.

That such a “quantization” exists is perhaps surprising. To a quantum system one expects to be able to associate a classical system by taking Planck’s constant to zero, but there is no good reason to expect that there should be a natural way of “quantizing” a classical system and getting a unique quantum system. Remarkably, we are able to do this for many classes of symplectic manifolds. For nilpotent groups like the Heisenberg group, that the orbit method works is a theorem, and this can be extended to solvable groups. What remains to be understood is what happens for reductive groups.

Already for the simplest case here, compact Lie groups, the situation is very interesting. Here co-adjoint orbits are things like flag manifolds, and the Borel-Weil-Bott theorem says that if an integrality condition is satisfied one gets the expected irreducible representations, sometimes in higher cohomology spaces. One can take “geometric quantization” here to be essentially “integration in K-theory”, realizing representations using solutions to the Dirac equation. Recently Freed-Hopkins-Teleman gave a beautiful construction that gives the inverse map, associating an orbit to a given representation.

For non-compact real forms of complex reductive groups, like SL(2,R), the situation is much trickier, with the unitary representations infinite dimensional. Vogan’s lectures were designed to lead up to and explain the still poorly understood problem of how to associate representations to nilpotent orbits of such groups. At the end of his slides, he gives two references one can consult to find out more about this.

Finally, there is a good graduate level textbook about the orbit method, Kirillov’s Lectures on the Orbit Method. For more about the orbit method philosophy, its history and current state, a good source to consult is Vogan’s review of this book in the Bulletin of the AMS.

I’ve just finished reading a wonderful new book by theoretical physicist Peter Freund, entitled A Passion for Discovery. Freund grew up in Romania, and began his career as a physicist in Europe during the 1950s, emigrating to the US during the 1960s, finally ending up at the University of Chicago, where he is now an emeritus professor.

When I was writing my own book I tried to include amidst the expository material about physics and mathematics stories of some of the people and events that seemed to me illustrative in one way or another. Freund has had the excellent idea of writing a book that foregrounds such stories, interspersing in the background the actual physics and mathematics. A reader who doesn’t know the science may not learn as much about it from this book as from others, but will get a feel for something perhaps more important, the “culture” of the field of theoretical physics. By this I mean the whole circle of knowledge that makes up the context in which theoretical physicists think and work. A reader who does know the science and some of the stories that Freund tells will deepen his or her knowledge by learning many more that he or she was probably unaware of.

When I moved from a physics environment to a mathematics one many years ago, one thing that struck me was that I had entered not just a field that studied somewhat different material, but a whole new cultural environment, very much like moving from the US to France. Different fields have different unspoken sets of values and beliefs, derived from their different environments and different histories. Shared stories about the history of the field and the quirks of leading figures of the subject make up a large part of this common culture. Freund does an excellent job of capturing the culture of twentieth-century theoretical physics, and one could learn much more about this from his book than from any textbook or most standard historical treatments.

It’s tempting to repeat here some of the stories that I learned from Freund’s book, but there really are too many to choose from, so I have to just recommend that you should read for yourself. Among the physicists you can learn about here are: Schrodinger, Heisenberg, Pauli, Dirac, Stueckleberg, Feynman, Salam, Chandrashekar, Zeldovich, Landau, Touschek, Thirring, Oppenheimer (who, unlike almost everyone else, comes off badly), Nambu, and many others. A significant number of mathematicians, including Emmy Noether and Andre Weil also put in an appearance.

Freund also does a masterful job of describing the story of how mathematics and physics operated under the totalitarian systems of the last century, including a description of how the Romanian dictator Ceausescu and his wife had the mathematics institute closed down and disbanded after their daughter, who was working there, spent the night in a resort motel with one of her colleagues. He tells the stories of some of the well-known German mathematicians and physicists who either collaborated with the Nazis or joined the Nazi party, and where this led their careers. There is also quite a bit about Russian physicists and mathematicians, illustrating their attempts to survive within the Stalinist system, and the institutionalized anti-Semitism that Pontryagin and others were responsible for supporting.

Freund describes particle theory research as generally having a single leading figure that the field follows. He sees 1905 to 1925 as the era of Einstein, 1926-1943 as that of Heisenberg, a transitional period led by Fermi, with Gell-Mann dominating from the fifties to the early seventies, at which point ‘t Hooft takes over, followed by Witten in the early eighties. Witten’s long era of dominance now appears to him to be coming to an end, and Freund nominates Maldacena as the leader for the new era which I guess has already been underway for a while, as AdS/CFT has dominated research for the last ten years.

While Freund is very strong on conveying the culture of particle theory that dominated the fifties, sixties, seventies and eighties, unfortunately he has much less of the same sort of material to help explain what has been going on for the last twenty years or so, the age of Witten and now Maldacena. There aren’t any stories he has to tell about Witten, ‘t Hooft, or any of the other researchers whose work has characterized this recent period. Perhaps part of the problem is that they’re a less entertaining lot: while I’ve heard a lot about Witten over the years, I can’t think of much in the way of really colorful stories.

Freund’s take on the current state of the subject is blandly optimistic: everything’s going just fine. He mentions the Landscape and suggests Susskind’s book for further reading, but doesn’t see a problem there other than that “we need time and perserverance”, and maybe cosmology will save the day. He does promote a more realistic point of view on the prospects for string theory, seeing it as a set of ideas that may in the future be part of some quite different real advance. His analogy is with Lagrangian and Hamiltonian mechanics, which didn’t really give anything you couldn’t get from Newtonian mechanics, but were necessary foundations for the truly revolutionary quantum theory.

All in all, Freund has written a fascinating book, one which any person who wants to understand more about the culture of theoretical physics can learn quite a lot from, whether they’re a novice to the field, or have spent much of their life in it.