Not Even Wrong

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.

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?

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.

I recently mentioned that funding for the RSVP experiment is being reevaluated. More details about this are available in a recent issue of Science magazine.

The Rare Symmetry Violating Processes (RSVP) project is a proposed experiment at Brookhaven that would have two components. One, “MECO” would search for neutrinoless conversion of a muon to an electron, observation of which would indicate new physics beyond the standard model. The other component, “KOPIO”, would try and measure the decay rate for neutral kaons to a pion, neutrino and anti-neutrino, a CP violating decay whose rate is predicted by the standard model.

Last fall the NSF had allocated money to start building the experiment, which was projected to cost $158 million. The idea was to use the AGS accelerator at Brookhaven, which in recent years has mainly been used as an injector for the heavy-ion collider RHIC. It seems though that revamping the AGS for use by RSVP may cost a lot more than people had originally thought, pushing the cost of RSVP up to as much as $300 million. The potential cost of RSVP is being reviewed, and HEPAP has been asked to evaluate the results that RSVP may be able to achieve at different levels of funding. According to Michael Turner, head of mathematical and physical sciences at the NSF, “We will reevaluate [RSVP’s] scientific value, its cost, and then make a decision.”

The SPIRES database is used each year to produce a list of the most frequently cited papers in particle physics. This year’s list has appeared, although the usual annual discussion of the list from Michael Peskin still hasn’t yet. The trends I commented on last year in the 2003 list are even more pronounced this year.

The top ten most highly-cited papers in particle physics are now dominated by experimental results in astrophysics and cosmology with five papers in this category. Particle theory is represented by three large extra dimension papers from 1998 and 1999, and a single string theory paper, Maldacena’s 1997 article on AdS/CFT. The Maldacena paper is now the fourth most highly cited particle physics paper of all time, surpassed only by citations of the Review of Particle Properties, Weinberg’s 1967 paper, and the 1973 Kobayashi-Maskawa paper.

Even more so than last year, this data shows that particle theory and string theory flat-lined around 1999, with a historically unprecedented lack of much in the way of new ideas ever since. Among the top 50 papers, the only particle theory ones written since 1999 are a paper about pentaquarks by Jaffe and Wilczek from 2003 at number 20, the KKLT flux vacua paper at number 29 and a 2002 paper on pp waves at number 32.

How many more years of this will it take before leaders of the particle theory community are willing to publicly admit that there’s a problem and start a diiscussion about what can be done about it?

For some other interesting statistical data gathered from this database, check out the SPIRES playground.

One relatively recent idea that probably hasn’t fully shown up yet in the yearly citation counts is Witten’s late 2003 idea about relating gauge theory and the topological string in twistor space. While the idea of working in twistor space has lead to a lot new results about gauge theory amplitudes, Witten’s original hope of relating gauge theory and string theory seems to be in trouble.

There’s a quite interesting discussion going on about Wick rotation over at Lubos Motl’s weblog.

In flat space-time, the situation is well-understood: if your Hamiltonian has good positivity properties you can analytically continue to imaginary values of time, and when you do this you end up with “Euclidean” path integrals, which actually make sense, unlike QFT path integrals expressed on Minkowski space, which don’t. You can see the problem even in free field theory: the propagator is given by an integral that goes through two poles, so is ill-defined. The correct way to define it to get causal propagation for a theory with positive energies is to go above one pole, below the other, which is equivalent to “Wick rotating” the integration contour 90 degrees to lie on the imaginary time axis.

In a curved space time, things are much trickier. And in a path integral approach to quantum gravity it is very tricky. Do you integrate over all metrics with Lorentz signature (ignoring the fact that the path integral doesn’t really make sense for a single one), or do you integrate over Euclidean signature metrics (Euclidean Quantum Gravity)? There are arguments against either choice, not to mention the non-renormalizability problems that both may have. For some of the arguments, see the debate in Lubos’s comment section, which gives some idea of how confused the state of this question is. Another good reference is the article by Gary Gibbons in the Hawking 60th birthday celebration volume. It doesn’t seem to be on-line, but his talk at the workshop is.

I’ve always thought this whole confusion is an important clue that there is something about the relation of QFT and geometry that we don’t understand. Things are even more confusing than just worrying about Minkowski vs. Euclidean metrics. To define spinors, we need not just a metric, but a spin connection. In Minkowski space this is a connection on a Spin(3,1)=SL(2,C) bundle, in Euclidean space on a Spin(4)=SU(2)xSU(2) bundle, and these are quite different things, with associated spinor fields with quite different properties. So the whole “Wick Rotation” question is very confusing even in flat space-time when one is dealing with spinors.

Over the years I’ve tried to sell the outrageous idea that one should define QFT in Euclidean space time, with one of the two SU(2)s in Spin(4) being Spin(3), the spatial rotations, the other being the SU(2) of the electroweak gauge group. I’ve never been able to get anyone to take this seriously, partly because I’ve never come up with a well-defined way of writing down path integrals which implement this idea.

There’s a story in this Sunday’s New York Times television section describing how Ed Witten pitched a story idea to the people who make the new TV show Numb3rs. According to one of the show’s executive producers, Cheryl Heuton, “Ed sent our script back along with an episode idea, which we used, telling us we should do something about a rogue mathematician who tried to crack Internet security by solving the Riemann hypothesis.” Witten had received the Numb3rs script to look at from his brother, the writer Matt Witten.

For more about the Caltech mathematicians who are the main consultants for the TV show, see this USA Today article.

Last night ABC News ran a two-hour primetime special on The UFO Phenomenon — Seeing is Believing. As part of this special program, they interviewed “one of the world’s leading theoretical physicists”, who, according to Bob Park, “looked a lot like Michio Kaku.” This physicist told ABC that UFOs should be taken seriously since “You simply cannot dismiss the possibility that some of these UFO sightings are actually sightings from some object created by … a civilization perhaps millions of years ahead of us in technology.” He also explained how aliens could get here using wormholes.

Kaku appeared yesterday on the radio show “Coast to Coast” to discuss UFOs and the ABC special. He appeared on the same show (in different hours) as Al Bielek, who evidently had a job in California, but regularly traveled by secret underground subway to Montauk, Long Island to work on the “Montauk Project”. During the 1980s he traveled to Mars on several occasions, as well as to “a research station in 100,000 BC, other planets to get canisters filled with Light and Dark Energy, and to the year 6037.”

I recently acquired a copy of the new volume 6 of Atiyah’s collected works, which contains things he wrote from the late eighties until very recently (the latest article is his joint paper with Graeme Segal on twisted K-theory). Unfortunately the price of this book is very high (about $200). I’ve bought cars for less than what I paid for the book.

Even more expensive is the full six-volume set, which Oxford intends to sell for $1000. Luckily I bought the previous 5 volumes quite a few years ago at a somewhat more modest price. Atiyah is one of my great heroes among mathematicians. He’s up there among the top very few in any reasonable list of the greatest mathematicians of the second half of the twentieth century, and the extent of his influence in bringing together mathematics and physics is hard to overestimate. Witten’s great work on topological quantum field theory was done very much because of impetus from Atiyah. One of the articles in the new volume is the write-up of Atiyah’s amazing talk at the Weyl Symposium in 1987, where he first suggested that there should be a four-dimensional QFT whose observables were Donaldson invariants and whose Hilbert space was Floer homology.

Atiyah is also known as Sir Michael. Before I heard about this I had always thought that the British system of honorary knighthoods was pretty silly, but the fact that they chose him gave me some respect for the whole system.

It’s a shame the books are so expensive, since they are wonderful documents that deserve wide distribution. Atiyah has not only discovered wonderful new mathematics, but he writes about it in an elegant, inspiring and lucid way. The books contain many expository pieces he has written over the course of his career, and these are pretty much all well worth reading. I regard a large part of my mathematical education as having come from spending a lot of time with these volumes over the years.