New This Week’s Finds

John Baez has just put out a new issue of his This Week’s Finds in Mathematical Physics, dealing partly in more detail with the material about Clifford modules mentioned here a couple weeks ago. I’ve added as the first comment here something he had some trouble submitting as a comment to the older posting on this topic.

John briefly mentions a relation of all this to Bott periodicity in topology, using a very abstract homotopy construction involving spectra. A more concrete version of this can be found in Milnor’s book on Morse theory. For the relation of Clifford algebras and K-theory, the standard refererence is the 1964 paper “Clifford Modules” by Atiyah, Bott and Shapiro published in the journal “Topology”. The crucial fact they describe is how the Thom isomorphism in K-theory (which is essentially the same fact as Bott periodicity) is related to the structure of Clifford modules. Greg Landweber has recently worked out an interesting equivariant version of this story.

Greg also has a nice new paper with Megumi Harada about the K-theory of a symplectic quotient, that looks like it should imminently appear on the arXiv.

John also mentions some recent work of Dror Bar-Natan, Thang Le and Dylan Thurston on the Duflo isomorphism. This is a beautiful story, and also has a relation to Clifford algebras that John doesn’t mention. For this, see Eckhard Meinrenken’s talk at the 2002 ICM in Beijing.

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Recent Conferences

Last week the 2005 International Linear Collider Workshop was held at SLAC, and the talks are available on-line. At the conference it was announced that Barry Barish of Caltech would lead the Global Design Effort for the International Linear Collider. The hope is to finish a design for the ILC in 2007, have a site chosen in 2008 and construction done by 2015, allowing the ILC to run at the same time as the LHC for several years, with each machine providing data that could help decide how best to use the other one.

This schedule seems overly optimistic to me. Because of the huge US deficits, getting the kinds of increases in the DOE budget needed to build the ILC in the US looks quite difficult, and, even if this were possible, funding constraints would probably stretch out the construction schedule. In Europe, CERN is devoting all its resources for a while to the LHC, and is backing an alternate, more speculative linear collider technology called CLIC. The most likely course of events seems to be that people will be waiting to see what the LHC finds and how the CLIC technology works out before fully committing to a new linear collider. If so, a decision about what to build and where to build it would probably not take place until almost 2010, with another decade probably required to actually construct the machine.

At the SLAC conference, the main theoretical talk was one by Savas Dimopoulos on New Models about his work with Arkani-Hamed on split supersymmetry and models where both the cosmological constant and the weak scale are anthropically determined aspects of the “Landscape”. There increasingly seems to be a disconnect between the experimentalists planning experiments at the LHC and ILC, whose plans often revolve around the search for low-energy supersymmetry, and the string theory inspired theorists, who are spending their time wandering around the landscape. From the landscape point of view, it seems that low-energy supersymmetry is extremely unlikely.

For more from theorists wandering around the landscape, see the talks at this week’s Workshop on N=1 Compactications at the Fields Institute in Toronto. The talks from this workshop are starting to become available on-line. Next week there will be even more of this at the Fields Institute, as it hosts a Workshop on String Phenomenology.

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Coleman Conference

I spent the last two days up in Cambridge, mainly attending the conference in honor of Sidney Coleman. Sadly, Coleman is in poor health, suffering from Parkinson’s disease, and was unfortunately unable to attend the talks in his honor. They were videotaped so that he could watch them later.

For me as for many particle theorists, taking Coleman’s quantum field theory course at Harvard was one of the great intellectual experiences of my life. Another such experience was reading and learning from his great Erice lectures, both the late seventies ones as they came out, as well as going back to his earlier ones that started in 1966. These were collected in 1985 in the book “Aspects of Symmetry”, allowing me and many others to replace a stack of dog-eared Xeroxes with a more durable volume. The fact that Coleman stopped giving these lectures after 1979 was to me one of the first indications that particle theory was entering a much less promising phase of its history. Coleman never really warmed to the topics of supersymmetry and string theory.

For much of his career Coleman played the role of guru for the particle theory community, generously sharing his unmatched insights into quantum field theory. He would sleep through the morning (famously announcing that he couldn’t teach a 9am class because he couldn’t stay up that late), get into his office late in the afternoon, then spend hours dealing with a long line of people waiting to talk to him to try and get some help with whatever problem they were working on. Steven Weinberg spoke for many people at the conference when he said that Coleman was the single person he had learned the most physics from.

The conference was extremely well-attended, with the large lecture hall in the physics building at Harvard overflowing on Saturday. I don’t think I’ve ever seen so many Nobel prize winning particle theorists in one place. They included Gell-Mann, Glashow, Weinberg, ‘t Hooft, Gross, Wilczek, Wilson, as well as Fields medalist Edward Witten. One of the few living Nobel particle theorists who couldn’t make it was David Politzer, who very much directly owes his prize to Coleman.

I won’t describe the talks in detail, this has been done pretty accurately already by Lubos Motl (who I got to meet in person for the first time). Physics weblogging was very well represented at the conference: besides Lubos, Jacques Distler was liveblogging from one corner of Science Center B on Friday, and Serkan Cabi was also there. Sean Carroll also has some comments about Coleman.

Among the more historical talks, perhaps the most interesting was that of Gerard ‘t Hooft (Lubos seems to have missed ‘t Hooft’s comment that he shouldn’t be referred to as “Gerardus”, a formal version of his name that appears on his Nobel citation and his passport, but is otherwise not much used). ‘t Hooft gave his version of the asymptotic freedom story. He said that he had computed the Yang-Mills beta function a couple years before Gross-Wilczek-Politzer, but didn’t realize that this result wasn’t known to the experts. He pointed out that everyone else had experience only in computing the scaling behavior of non-asymptotically free theories, whereas the first theory he did the computation for was an asymptotically free one, so he thought this was unremarkable. He did say that Gross-Wilczek-Politzer deserved the Nobel since (besides being the ones to publish the beta-function result) they had understood how to use this to explain Bjorken scaling, something that he hadn’t known about. He said his advisor Martin Veltman had told him that the Yang-Mills scaling behavior wasn’t relevant to experiment since experimentalists only cared about what happens on mass-shell. Luckily Veltman was one of the few Nobel theorists not in attendance, since he would likely have blown a gasket if he had been there to hear some of the things ‘t Hooft had to say about him. ‘t Hooft went on to say that he had learned one important thing from this episode: always immediately publish any new result you have.

There was significant mention of string theory in only two talks, those of Gross and Witten. Gross gave essentially the same talk he gave last October at the 25th anniversary of the KITP. He joked that he had managed to time the award of the Nobel with the KITP celebration by every year for the last thirty years writing to the Nobel Committee and asking them to wait a while before awarding him the prize, something they had been happy to do. At the point of his talk when he said that the question “What is String Theory” was one of the big questions for the future, he stopped to defensively note that since we don’t know what string theory is, it is an idea that can’t be killed, no matter how much certain members of the audience wanted to do this. He went on to claim that since AdS/CFT kind of connects string theory with QCD, string theory is in some sense part of the standard model, so it’s importance is secure. This argument seemed to me pretty disingenuous, since presumably he’s well aware that the problem most critics have with string theory is not with the idea of using it as a dual representation of QCD, but with the idea of getting a TOE out of it, a project which some have called a “colossal failure”. He didn’t have anything to say either about this failure or about the whole Landscape mania.

The last talk was Witten’s, entitled “Emergent Phenomena in Condensed Matter and Particle Physics”. He started by saying that he was afraid the title of his talk might be more exciting than the talk itself. By “emergent phenomena” he meant roughly non-perturbative phenomena in QFT, where the long distance degrees of freedom one observes are not directly related to the local degrees of freedom. He gave QED as an example of a non-emergent theory, QCD an emergent one, with the nature of the electroweak theory still up in the air until we know more about the origin of electroweak symmetry breaking.

He went on to say that gravity messes up this distinction between local and emergent phenomena, since one doesn’t have diffeomorphism invariant local observables. He then quoted his 1980 work with Weinberg (and with help from Coleman) to the effect that you can’t get a massless spin two bound state in a theory with a local stress-energy tensor, saying that this showed that you can’t start with a local theory in Minkowksi space and generate Einstein gravity as an emergent phenomenon. For him the lesson is that if you want gravity as an emergent phenomenon, you need to find a way to first get space-time as an emergent phenomenon, and he believes that whatever the primoridial M-theory underlying string theory is, it should do this. While such a theory doesn’t now exist, he went on to give the AdS/CFT correspondence as the kind of thing he had in mind. There the Weinberg-Witten argument is evaded since a QFT in 4 dimensions is related to a gravity theory in a different number of dimensions (5).

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Anniversary

It’s now been exactly one year since I first set up this weblog. At the time I thought the number of those sharing my interests would be very small and hardly anyone would be looking at whatever I put up here. Things have turned out very differently, with an ever increasing number of connections. I started gathering statistics in May of last year. Here’s the average number of connections to the main page per day (there’s a similar number of connections to other pages, from Google searches and links from elsewhere).

May 2004 146
June 2004 240
July 2004 281
August 2004 336
September 2004 315
October 2004 514
November 2004 514
December 2004 572
January 2004 735
February 2005 955
March 2005 (first half) 1109

I’ve enjoyed and learned a lot from many of the comments posted here (at last count there have been 2728, concerning 168 different postings), but a recurring problem has been that many people would like to turn the comment section into a discussion forum for their own personal speculative ideas about physics. This threatens to completely overwhelm discussion of the topics I’m actually posting about. This morning I had to delete several several such comments from different people. Please do not post comments here of this kind. Get your own weblog and do it there. If every so often you want to post a link here to something you’ve written of this kind elsewhere, that’s fine.

It’s been quite a year, especially as the story of string theory just gets weirder and weirder. I have no idea what will happen during the next year, but I’m looking forward to finding out.

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No Cosmological Constant?

A paper appeared on the arXiv last night entitled Primordial Inflation Explains Why the Universe is Accelerating Today by Rocky Kolb of Fermilab, together with Sabino Matarrese, Alessio Notari and Antonio Riotto. There’s also a Fermilab press release about it today.

I’m no expert on the subject, and would love to hear the opinion of someone who is. As near as I can figure out the idea is that what is really responsible for the effects that have been ascribed to a cosmological constant is a “cosmological perturbation” of the gravitational field. This is supposed to be a perturbation that expanded during the inflationary period so that its wavelength is now larger than the Hubble radius. According to the authors, this predicts a different magnitude vs. red-shift relation than the standard cosmological constant does, so their idea should in principle be testable.

If they’re right, this certainly will cause a huge problem for the whole “Landscape” business, which has advertised as its greatest success the “prediction” of a non-zero cosmological constant of the right order of magnitude.

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Skeptical SF Chronicle Article

Today’s San Francisco Chronicle contains an article about string theory entitled “Theory of Everything” Tying Researchers Up In Knots. It’s by science writer Keay Davidson, and is about the most skeptical article on string theory I’ve seen in the mainstream press. The lead sentence is:

“The most celebrated theory in modern physics faces increasing attacks from skeptics who fear it has lured a generation of researchers down an intellectual dead end.”

Davidson contrasts Michio Kaku’s very pro-string theory point of view in his new book Parallel Worlds, with the much more skeptical views of Lawrence Krauss, who evidently has a book entitled “Hiding in the Mirror: The Mysterious Allure of Extra Dimensions” coming out in September. He also got comments about the current state of string theory from quite a few different people, including yours truly. The article contains a link to this weblog.

Some of the string theory critics quoted are just inherently opposed to any new mathematical approach to fundamental physics, something I have no sympathy with. One of these is Stanford’s Robert Laughlin, who makes the point that string theorists are trying to camouflage the theory’s increasingly obvious flaws by comparing the theory to “a 50-year-old woman wearing way too much lipstick.” Because of Laughlin’s extreme anti-mathematical theory views on the one side and those of his colleagues like Lenny Susskind on the other, “The physics department at Stanford effectively fissioned over this issue” says Laughlin. He goes on to say “I think string theory is textbook ‘post-modernism’ (and) fueled by irresponsible expenditures of money.” For the record, I’m no more of a fan of Laughlin’s views about particle theory than I am of Susskind’s.

Some of the quotes from defenders of string theory are a bit strange, with none of them addressing the fundamental problem the theory is facing these days as it becomes obvious that it can’t predict anything. John Schwarz is quoted as saying “string theory is the only approach that has the potential for explaining dark energy” which is kind of peculiar since it is well-known that superstring theory naturally leads one to expect a value for this energy density that is off by 120 orders of magnitude. The only way around this seems to be the “landscape” argument, in which you essentially give up any hope of ever predicting anything. The other defenders of string theory quoted in the article mainly try and claim that twenty years of work on the theory is still nowhere near enough, that it is way too early to be able to evaluate it yet. They don’t give any indication of how much longer we should wait for such an evaluation, but if twenty years isn’t long enough, it sounds like they hope this won’t occur while they’re still alive.

Update: For a very different take on this, see Lubos Motl’s posting.

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Clifford Modules

John Baez had a weblog long before the term was even invented, and for many years now has been consistently putting out interesting current material about math and physics under the title This Week’s Finds in Mathematical Physics. The latest edition has a beautiful explanation of the structure of modules of the Clifford algebra.

Traditionally one thinks about geometry in n-dimensions in terms of n-dimensional vectors and tensors built by taking tensor products of vectors. These are all representations of the general linear group GL(n), or if one has a metric, the othogonal group SO(n) of transformations that preserve the metric. However, it turns out that there are representations more fundamental than vectors, the spinor representations. These require a metric for their definition, and are projective representations of SO(n), or true representations of the double-cover Spin(n). When one tries to construct spinors, one quickly runs into a fundamental algebraic structure associated with a real n-dimensional vector space: the Clifford algebra C(n). Spinors occur as “modules” of the Clifford algebra, i.e. vector spaces that the Clifford algebra acts on. The structure of these possible Clifford modules is rather intricate, with a certain eight-fold periodicity. Baez gives a beautiful explanation of part of this story.

Physicists generally complexify everything in sight (i.e. assume all numbers are complex), which makes things much simpler. Then the story is periodic with period 2 instead of 8, and Clifford algebras are just one or two copies of a complex matrix algebra of k by k matrices, where k is some power of 2. Clifford modules (including the spinors) in this case are just complex vector spaces of dimension k, and tensors built out of these. One good place to read about all this, together with its relation to the index theorem, is in the book “Spin Geometry” by Lawson and Michelson, but there are by now lots of others.

If one believes in a deep relation between physics and geometry, these Clifford modules should somehow come into play in the structure of the most fundamental physical theories. To some extent this is already in evidence in the way spinors and the Dirac operator occur in the standard model. There are also tantalizing relations between the idea of supersymmetry and the Clifford algebra story. Many, many people have been motivated by this kind of idea over the years to try and use Clifford algebras to come up with a fundamental particle theory, one that would explain the structure of the standard model. While some of these attempts have very interesting features, none of them yet seems to me to have gotten to the heart of the matter and used this kind of geometry to give a really convincing explanation of how it is related to the standard model. Some crucial idea still seems to be missing.

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Brane Damage at Fermilab

Last week Shamit Kachru gave a colloquium at Fermilab with the title String Theory and Cosmology. The scariest part was the beginning when he noted that what he would be talking about was work due to 500-1000 theorists and he put up a couple slides listing many of them.

He spent the first part of his talk laying out the “Landscape” story, somehow neglecting to mention that it was ugly, completely unpredictive, and told us nothing at all about the properties of the world today. He then moved on to discuss branes and cosmology, not making clear that branes explain absolutely nothing about the early universe or cosmology, although they do give you a new slogan he has come up with:

“Big bang as brane damage”

There were a couple questions at the end, with no one standing up and asking if this was a bad joke or something. I’m curious if anyone from Fermilab can explain to me what a typical experimentalist’s reaction is to this kind of talk:

1. Are they impressed by this stuff and don’t realize they’ve been fed a load of pointless nonsense for an hour?

2. Are they smart enough to realize they’ve just sat through an hour of pointless nonsense, but are too polite to say anything about this at the end of the talk?

3. Are they so smart they know in advance this will be an hour of pointless nonsense, so don’t even attend, and are off somewhere else getting real work done?

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Hans Bethe 1906-2005

Hans Bethe died at home in Ithaca, New York on Sunday. There’s an extensive obituary in the New York Times.

I believe Bethe was the last remaining figure still alive from the generation of physicists who came of age with the new quantum mechanics during the mid-to-late 1920s. Some popular lectures on the topic of “Quantum Physics Made Relatively Simple” that he gave for his neighbors in 1999 are available on-line.

Update: There’s more about Bethe and Cornell at Matthew Nobes’s weblog.

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Higgs Search at the Tevatron

Tommaso Dorigo of the CDF collaboration at the Tevatron has just posted (with commentary), the slides for his talk at Moriond later this month about the status of the search for the Higgs at the Tevatron. The bottom line is that with the data they have already analyzed they are still quite a ways from being able to see the Higgs, but, if its mass is just above the lower limit set by LEP2, they should be able to see it by two years from now. With quite optimistic assumptions about the performance of the Tevatron, by the end of 2009 they should be able to see the Higgs if its mass is less than 180 Gev. He ends by saying that at “95% confidence level” he thinks the Tevatron will be able to end up seeing a Higgs up to 135 Gev mass, and if its mass is just above the LEP2 limit at 115 Gev, they should have 3 sigma evidence for its existence.

By 2009, the LHC should be producing data and putting the Tevatron out of the Higgs discovery business. For a bewilderingly complicated schedule of the LHC construction and installation, go here. From what I can tell, they are still on track for first colliding beams in spring of 2007.

Update: See Tommaso’s comment to this posting for a clarification. By “seeing the Higgs” I didn’t mean to imply that they would be able to prove the Higgs was there, just that they would be starting to see some evidence of its existence.

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