A few years ago various US universities decided it was a good idea to offer a course on string theory for undergraduates (see here), but in recent years most of these seem to have been dropped from the curriculum. Brown University is going in the other direction, offering Physics 1970C, String Theory for Undergraduates, this semester. A report from a Brown undergraduate on Lubos’s blog gives me some encouragement to continue blogging:

Life is carrying on naturally. In fact, if I hadn’t been reading eg woit’s blog, I would’ve suspected we’re still in the middle of a stringy revolution! We even just started a new string theory course for undergrads, and I and quite a few other undergrads held a string theory seminar. Interest in stuff like LQG is completely zero. So in the press you have woit, smolin blahblahblahing, ok. But in the meantime, you have ads/cft, and the whole twistor reformulation of yang mills in terms of contour integrals over grassmannians (inspired by twistor string theory). Even condensed matter physicists accept string theory as one of the greatest things that happened to physics.

The undergraduates at Brown have a String Theory Study Group, Facebook group here.

**Update**: The undergraduate string theory courses are now facing some competition. LSU is offering an undergraduate Introduction to Loop Quantum Gravity.

I am not a great fan of superstring theory but I am also not sure this course is a bad idea.

I think it is good for students to be exposed to the fact that Physics is an evolving lively subject that is often controversial.

How about a course with basic ideas from three different approaches to quantum gravity – LQT, strings and maybe Dynamic triangulation ?

Lets face it – the standard model is -well- pretty standard. I think there should be some courses in a Physics degree that excite and risk being wrong.

Paul,

The main problem with teaching undergraduates about proposals for how to quantize classical GR is that to even understand what the problem is, they first need to understand classical GR as well as how to quantize a classical field theory. The number of undergraduates who understand GR and QFT is quite small, and they would be best off in graduate courses.

Encouraging undergraduates to do coursework on the various controversies over quantum gravity that haven’t led anywhere interesting in recent years, before they have the background necessary to appreciate the crucial technical issues, seems to me to be a bad idea.

I’ve been a long time reader of this blog, this is my first comment.

I’m in this class! I’m a senior at brown. The class about 5 undergrads and 3 grad students, taught primarily by a postdoc (Ari Pakman) and by prof. Antal Jevicki when he is available.

I’ll answer anything you want about it, if you have further curiosities. I’m personally rather skeptical of string theory , but only based on what I consider to be second-hand information. I’ve never sat down and tried to understand the maths. I thought this class would be good to not only close that gap.

Where did you find that quote?

Max,

The quote is from “gigel”, commenting at Lubos’s blog on his recent posting about “expanding crackpottery” that I blogged about here.

Good luck with the course, I’m sure you’ll learn something from it (although I think students would be much better off learning quite a bit of quantum field theory before tackling string theory).

The only real textbook I know of at this level is Zwiebach’s, so I’m a bit curious to hear if some other source is being used as a text for the course.

The source is indeed the zwiebach book. I’m sure that most people in the class are going to have holes in their physics fundamentals, but it is being taught in a manner that will hopefully have a chance to address those shortcomings. I don’t expect to have a robust understanding of string theory or ads/cft at the end of the semester, but I’m excited for the chance to learn first-hand what everyone is getting so worked up about; I might not get another chance.

The problem I see with trying to teach AdS/CFT at this level is that students don’t even know what the CFT is, and giving them that kind of background is not going to be possible in a course like this.

AdS/CFT is an interesting topic, but, like much of string theory, it is surrounded by vast amounts of hype. Enjoy, but don’t believe everything you hear…

Hi Peter,

I think quite a few of the undergrads in the course do know both qft and gr, so this seemed like a decent opportunity to build on both.

You can’t deny that string theory has been pretty useful in all sorts of places. I don’t even think that the whole unification bit is the most amazing (possible) feature of string theory. You can take it as a tool to solve condensed matter problems, or just a new framework to test your undergrad knoweldge of lagrangians. I’m not sure where doubt has any room in this.

The two topics in the quote above are just a few examples where string theory did prove useful. Can you possibly deny this in any way?

gigel,

On its main selling point (unification), string theory has been an abysmal failure, something which Zwiebach’s book doesn’t really acknowledge.

Sure, string theory is an interesting exercise in how to quantize an infinite dimensional classical system. The problem is that it is really a quite difficult exercise (due to the infinite-dimensional invariances of the system). Zwiebach covers some of this, but it seems to me that most people’s time would be better spent first understanding more physically relevant QFT examples. If Brown undergrads really do understand QFT at the appropriate level, they should be ready for a graduate-level course.

The use of AdS/CFT as a tool to solve condensed matter systems is a hot research topic, but it remains very unclear how powerful it is. Your claim that “condensed matter physicists accept string theory as one of the greatest things that happened to physics” is pure hype and nonsense.

String theory as a tool to solve condensed matter systems? As far as I know, this topic has become hot among particle theorists, but has not had any real impact on condensed matter physics. Until condensed matter physicists think AdS/CFT has solved problems that they can’t solve before, and until AdS/CFT becomes an essential part of condensed matter physicists’ tool set, this is just another speculative research direction.

Ok I was obviously hyping that part.

Peter, people sometimes are eager to see research or more “fun” topics before they reach 30 and complete their phd. Following the standard courses very often drives people away from physics. Finding about Noether’s theorem in grad school? Finishing your SB in physics and not knowing what the weak and strong forces are (ie, about half the forces in physics)?

It happened that many undergrads here were interested in the topic, so the university offered the course. No one attempted to brainwash us or drag us into the bottomless pit of string theory research. It was student requested.

Slightly off topic, but Peter, how is your research going on BRST?

You said something like a year ago that by the summer of 2009 you will have publishable results, at least a preprent, what happened to that?

Cheers,

Trent

It’s certainly good when people try to learn about the issues. On the other hand, learning what string theory

is(while difficult enough) is not the same as getting the right background to assess its prospects. For that one needs understanding of general relativity and quantum field theory at a deep — really, a foundational — level.Very few physicists work at such a level. There are many really excellent physicists, but virtually all work is done on problems which are solved by technical acumen within existing frameworks, or modestly extended ones. So it is very hard to point a student to someone or some reference which would help with a critical foundational assessment of string theory (beyond the semi-popular literature).

I should point out that a main tenet of string theory has been that gravity

canbe quantized by a modest extension of existing framework (getting a quantum field theory over strings, rather than over points). (This is not “modest” within the class of quantum field theories, but it is extremely modest from the point of view of what might potentially be involved in reconciling quantum theory and gravity. That might be something entirely different from quantum field theory.)I second Trent’s concern. I do hope Dr Woit will publish more papers. Both this blog and the related book have been excellent, and the world has already received the message. Maybe it is a good time to move on.

Pawl, you are forgetting about holography. It might turn out that the better definition of string theory and quantum gravity comes from ads/cft. There are many possibilities, not just the old “everything is made of tiny vibrating strings singing along”.

Whichever way you put, ads/cft is just to remarkable to brush off. At least that’s what I think. Typical unification pales in comparison.

I mean, on the one hand, what, everything is not made from point particles, but from strings. ok. On the other, you have two completely different theories living in different spaces and being equivalent. That’s the real wow.

Gigel,

As duality between quantum field theories, holography is interesting; as a statement about gravity, is on far shakier ground.

In fact, the history of the subject is something of an object-lesson in the point I was making. It was originally introduced by particle theorists who wanted to regard black holes as just another species of particle — and showed little awareness of the differences in the causal structure of black-hole space-times from others. This led to misunderstandings which persist to this day, and also made the physical basis of the theory fuzzy.

Its recent manifestations are still rather fuzzy. (There is, for instance, the over-arching question of what the entropy really is and whether one can meaningfully speak of the entropy of a general gravitational configuration. There are also serious technical problems.)

Some good might come out of it, but what I am arguing for here is the necessity of an assessment which fairly faces up to the problems. Its proponents would do well to demonstrate their own critical awareness by bringing these forward.

And besides, the theory really encourages you to look at many different topics, including in mathematics. I’m not sure why anyone would call that “wasting time”. Mathematics is useful to anyone studying anything, and I mean anything, not necessarily science.

I’m pretty sure you can put all the math and techniques to good use in other places if string theory were to somehow fail (though there’s no way that could happen since it’s intrinsically related to YM, in at least two vastly different ways).

Pawl, I’m a very big fan of being critical and pointing fingers at the problems. But no one is hiding them, like some people would say. I mean, most papers have a dedicated section at the end on remaining open problems. In fact, that’s probably everyone’s favorite section of a paper. What more would you want?

Sure, you probably won’t hear what the difficulties of ads/cft are in popular press, if that’s what you’re referring to. But people do know about them. And how do you get people to solve them? You obviously don’t hold a talk and start with “so we have this thingy, but man it has so many problems! so, any volunteers to help us out?”. You would probably want to say something like “ok we have this thing, we could use it for this and this and this. who wants to develop it further?” Notice the difference?

I think this is pretty much valid in all fields of science. Think of eg quantum computing, which would be the extreme example.

The “publish or perish” thing probably isn’t helping much either. If people need to put out a few papers per year, they don’t really have much time left to tackle the truly formidable conceptual problems. A “we rederive the relations of aaa et al using absolutely nothing new” still sounds better than “ya, i thought really hard about this for a couple of years, I even asked some friends, but didn’t get anything. sorry. next time, I promise”. But of course everyone knows about this. Not an easy to solve issue.

In the meantime, what’s wrong with some undergrads having fun while they still can đ . Either they’ll like it and be happy to have found something worth their time and effort, or they’ll smell something fishy and cross out a potential research topic early on. It’s not like we’re learning something completely useless. The majority of people in het are still into strings, so it’s sensible to prepare undergrads for the outside world accordingly. As in absolutely any other field.

On a slight tangent, are there any universities which officially offer a regular undergraduate course on quantum field theory?

I suppose if such an undergraduate course on qft is offered, the easiest way to do it would be to do canonical quantization of the scalar fields with tree level and 1-loop calculations of phi^3 interactions in 6 dimensions.

Gigel,

You bring up a number of points.

(1) Sure, mathematics is useful. It’s not the same as foundational physics, however.

(2) You want to be “a big fan of being critical,” but you also would, it seems, view it as a surprise if string theory “somehow turned out to be wrong;” you cite connections with Yang-Mills. This is mixing mathematical results and fundamental physics. The math is reasonably secure; the idea that we have reason to be less than skeptical of string theory as a fundamental physical theory is not supportable.

I really have to underline how gross this is. It is extremely hard to get physics right in (for example) an energy regime which is half an order of magnitude beyond what has been experimentally investigated; there is, for example, no consensus at all on what results will emerge from the LHC. We’re usually simply not clever, or imaginative, enough to guess; nature surprises us. If (as in string theory) one aims to describe the Planck scale — well, it’s good to be ambitious, but simple experience shows that it is far more likely one will get it wrong — and seriously wrong — than right. Any claims of naturality of beauty that string theory has are no defense here: beautiful theories are (to borrow a phrase of Haldane’s) slain by ugly facts all the time.

I’ve limited myself to generalities which would apply to

anyspeculative fundamental theory, but I could also list specific reasons for being especially dubious about string theory.(3) Yes, papers mention open problems. It’s the problems they

don’tmention which I’m concerned about. I don’t think the authors have any conscious intent to conceal; it’s rather that they’re unaware of the problems, or of how serious they are.(4) I was not referring to difficulties with ADS/CFT, but to two other things. The first were the original arguments (ca. 1993?) about holography and black holes, which depended heavily on supposing CPT invariance held but didn’t take into account that the time-reverse of a black hole is a white hole. (This mistake is still being made in somewhat different contexts.) The second was the cavalier assignment of entropy to regions in space-time.

(5) It’s important to be quite frank with students (whether research students or undergraduates) about the chances of success (in the sense of progress on fundamental physics), and not just to be a cheerleader.

No study of string theory is complete without reading the introduction to hep-th/0204131, in particular subsection 1.6. And keep in mind that the author is the founder of the string theory group at Rutgers.

With reference to “some encouragement to continue blogging”, I follow your blog and Jester’s for the great “executive summaries” you both provide of what’s going on in high energy physics.

Peter,

I take you point – but how are you going to attract people to people with just the same old courses on Electrodynamics, Optics and Thermodynamics ?

After my first degree i had no exposure to GR, weak interaction etc other than what I read for myself.

I think that there is a need to compromise academic rigor here against the need to give people some education about what the cutting edge of Physics is about.

Otherwise people might end up going into Engineering or (gulp) even Biology…

I get frustrated when I see Physics taught as if it were Ancient Greek rather than a lively active discipline.

[ sorry meant people to Physics of course]

Trent, petergreat,

What I’m working on is still the idea outlined in toy-model form in

http://www.math.columbia.edu/~woit/brstdirac.pdf

I haven’t finished that paper since I’m still not quite clear on where it is going and I want to understand this better before finishing it. The next step is to understand the case of affine Lie algebras, and the relationship to what people have done in geometric Langlands. I’ve been learning a lot about this, especially about the D-module approach to representation theory, and this has given me a new perspective on this material. I’m quite excited about this, but still in the middle of it, hope to have an updated and maybe final version of the paper this spring, we’ll see…

gigel/Paul,

I’m in no way opposed to teaching material beyond 19th century physics to undergraduates. If they graduate without being exposed to some version of Noether’s theorem (i.e. that symmetries imply existence of notions of energy, momentum, angular momentum, charge… and their conservation), that’s educational malpractice.

Also not against fun. There are a lot of ways though to choose interesting and challenging material about modern physics to try and explain, without picking a 40 year old set of speculative ideas that haven’t worked out and are in the process of being abandoned by the community. If you want to present something cutting-edge that is going to get ambitious undergrads a jump on hot topics in current research, string theory is a peculiar choice these days.

It might be fun and useful to have a course on

“what is screwed up with the Standard Model”.

Pawl,

It would be very hard to believe that all the results from ads/cft and twistor string theory are just mathematical coincidences. It is much more likely there is something going on which we still have to uncover completely. The fact that ads/cft is a complete equivalence really does say you can’t kill string theory.

But also, you can’t really make the distinction between “mathematical” connection and “physical” connection as you suggest. The Standard Model is also just a mathematical connection with nature. It’s a model, a translation of nature to our own language. It’s good because some calculations you do with it turn out correct. String theory might be a different translation of Yang Mills. In one of its manifestations.

Gigel,

You seem determined not to distinguish between mathematical and physical successes. While physics is indeed cast in terms of mathematics, there is a huge difference. Physics makes testable connections with the world in ways which are as precise as possible. String theory — as a fundamental physical theory — has no successes at all in this regard.

The position you have been taking is indeed one which has at times existed within the string-theory community. However, it amounts to an abandonment of the basic goals of physics, and the motivation for this radical step is nothing more than aesthetic attachment to a “theory” with no phenomenological successes!

Pawl, Gigel,

Please, enough of the tired pro/anti-string argumentation, unless someone has something new to say.

I don’t know what other advanced elective courses Rutgers offers, so it’s hard to put this in context.

As a physics professor in a small department, I occasionally get students coming to me expressing interest in string theory. Tellingly, they are rather intellectually incurious about the steps before they get to string theory. They’ve heard of GR, maybe even taken a class in it, but they haven’t really heard of quantum field theory (due to budget cuts we can’t offer such a course), and they certainly haven’t heard of gauged field theories. And they seem completely uninterested in such topics; they seem to view gauged field theories, not as the theoretical triumph that they or, not as a rich field of ongoing investigation, but as a pesky inconvenience on their way to becoming the next Einstein. There’s no real intellectual curiosity; they just want to be the next Einstein.

I can see string theory (or LQG) as being part of a more general survey course, which would include QFT, the standard model and gauged field theories, and ending with quantum theories of gravity. And if Rutgers or LSU do this but call it a “string theory/LQG for undergraduates” to draw in more students, more power to them.

CWJ,

This has been going on for more than 20 years. Back when I was in grad school, it was common to find grad students (and some highly motivated undergrads) who had the mentality that quantum field theory, gauge theories, classical GR, etc … were pesky “inconveniences” in their quests of being the next Einstein. I remember a few students who didn’t even bother taking the quantum field theory courses, and went straight to studying stuff like string theory papers (various Schwarz lectures), and later the Green, Schwarz, Witten superstring theory books.

Well Peter, my undergraduate degree didn’t cover Noether’s theorem. The address is University of Cambridge, Cambridge, U.K if you want to sue them for educational malpractice :-).

I think universities in general do a lousy job of keeping the interest of their students in Physics. I don’t believe final year undergraduates can’t cope with the basics of GR, Higgs mechanism etc. The details can wait for grad school.

Most undergraduates don’t become Physics professors. I think it is a disservice not to give them at least a flavor of Physics developments after 1915. Maybe things have improved since 1985 when I graduated but I doubt it.

By the way I think the (well-deserved) success of your book and others like it show there is a huge demand for this kind of course. I know Feynman tried it with QED. Maybe not strings but at least a few Feynman diagrams and an outline of the standard model.

Paul Wells,

Several of my former colleagues have taught undergraduate particle physics courses using the Griffiths textbook on elementary particles. Their courses typically covered tree level calculations in the Standard Model, along with whatever background stuff that was needed to understand such calculations (ie. Higgs mechanism, SU(2) and SU(3) groups, Dirac equation, etc …). They didn’t bother going extensively into renormalization, other than the simple minded approach taken in the Griffiths book.

In the end, it was essentially a course in calculating Feynman diagrams by being given the Feynman rules, without going through much of the quantum field theory formalism. Not entirely satisfactory, but it was the least they can do without making the course too difficult.

Paul,

You may be talking about Cambridge specifically as the Physics course you did is embedded in the Natural Sciences Tripos which -even if you take all the physics options – has a lot of non-physics stuff in it, giving you less time to (e.g.) study the Higgs mechanism. In the first year of any physics course one has to do a lot of calculus, linear algebra and other mathematics, plus thermodynamics, electromagnetism and classical mechanics. I don’t see how one can avoid that. In the second year, one needs to study quantum mechanics, atomic physics, nuclear physics, statistical mechanics, solid state physics, special relativity, more electromagnetism and optics. Again, I don’t see how one can avoid that. So – maybe in the third year, bearing in mind that all the lab work will be a commitment throughout – one could get more up-to-date. At Oxford the particle physics 3rd year option (at least in 1980) got us pretty up-to-date, but it was an experimentalists course as relativistic QM and QFT were not taught. Neither was GR. I am in two minds about teaching GR to undergraduates, but certainly think that the theory of continuous groups should be taught – and preferably before even one starts on quantum mechanics. One should know what SU(2) is before studying angular momentum in QM. Noether’s theorem? I am not sure. That would have to be embedded in a QFT course – and to do that in sufficient depth would require cutting out a lot of important other stuff.

Peter, I can’t believe you said this:

”

Please, enough of the tired pro/anti-string argumentation, unless someone has something new to say.

”

What?!? Using this argument you could close your blog!

If you mean the above genuinely why don’t you fold the tent?

I’m kinda confused.

Trent

Noether’s theorem appears first in classical mechanics. It can be explained in 5 to 10 minutes to students who already have studied cyclic coordinates and their relation to conservation theorems. Proving it requires only a chain-rule differentiation and takes just one line.

I’ve always wondered why textbooks like Goldstein or Landau and Lifshitz do not even mention Noether’s theorem, whereas they spend a lot of space on much more abstruse topics with many fewer practical applications.

anon,

re: Noether’s theorem

We were given Noether’s theorem as a homework problem in an undergraduate Lagrangian mechanics course.

Chris Oakley,

Maybe I should have went to Cambridge for undergrad. đ Back when I was an undergrad, it felt frustrating not doing any modern physics stuff earlier in the curriculum.

For example, we didn’t do any quantum mechanics until the tail end of junior year (ie. third year). That junior year quantum course basically covered modern physics topics (ie. Planck, de Broglie, Bohr, etc …) and some simple solutions to the Schroedinger equation (ie. free particle, square well, Heisenberg inequality, etc …) at the end of the course.

For most of the first three years of physics undergrad, it was largely classical physics done in successive courses with more mathematics added in each year. So by the time we took that first quantum course in junior year, we already had taken: 2 courses in classical mechanics (ie. with and without Lagrangian mechanics), 1 course on electromagnetic theory (ie. solving Laplace’s equation), 1 course on complex analysis (ie. resides, Cauchy’s theorem, etc …), 1 course on partial differential equations, 1 course on physical optics (ie. solving Maxwell’s equations, etc …), and 1 course on classical thermodynamics + heat transfer (without any statistical mechanics).

I suppose learning all that mathematics and classical physics stuff in extensive detail before ever doing quantum mechanics, does take some of the “deus ex machina” out of quantum mechanics. (For example, the analogy between Poisson brackets and the commutators of operators in quantum mechanics). But unfortunately it didn’t leave much time for the more modern courses like particle physics, nuclear physics, solid state, etc … By the time we were able to take the particle and/or nuclear physics courses in senior year (4th year), it was the last semester before we graduated. We all had a bad case “senioritis” by then. đ

I remember several former classmates I knew in my freshman undergraduate year who decided to transfer to another university, because our university didn’t do any “cool” modern physics stuff in sophomore or junior year (2nd and 3rd years). They were very impatient for the most part, and wanted to do quantum mechanics early on. Years later I found out a few of these same former classmates who transferred out, were also into string theory years later. They were the same ones who thought that it was a brilliant idea to skip the quantum field theory and general relativity courses, and go straight to reading string theory papers.

I ended up taking the particle physics course (along with the nuclear course) in the last semester of senior year. It was largely an “experimentalist” course, where quantum field theory and relativistic quantum mechanics were not covered at all. At the time it seemed kind of disappointing.

“If they graduate without being exposed to some version of Noetherâs theorem (i.e. that symmetries imply existence of notions of energy, momentum, angular momentum, chargeâŠ and their conservation), thatâs educational malpractice.”

Well, I didn’t see that as an undegrad; had to wait to become a grad student to see Noether’s theorem.

Peter, on the BRST point: You said here, and I’ve seen a few other references that seem to track your language, something along the lines of, that BRST “comes close” to explaining why there are fermions. Could you explain what you mean by that? I’ve been trying to understand it, but its clearly far deeper into the math than I’ve been able to penetrate.

@JC says: “On a slight tangent, are there any universities which officially offer a regular undergraduate course on quantum field theory?”

IIRC, Caltech offers Ph205 (?) which I’m told is basically a QFT course.

Undergrad physics majors who stayed the course — ie, didn’t change majors halfway through their years — and kept pace on their math courses, would be taking Ph205 in 4th year.

Ph205 was nominally a graduate course, but beginning graduate courses were also populated with 3rd/4th year undergrads.

Tom,

When I was in grad school, it was not unusual for highly motivated undergrads to take graduate level courses. When I took quantum field theory in grad school, there were several undergrads enrolled.

With that being said, what I was asking in my previous question is whether there was an undergraduate course on quantum field theory which was not cross-listed with the graduate qft course.

Now that I thought about it more, I suppose such an “undergrad qft” course could in principle cover quantum electrodynamics done using canonical quantization. What I had in mind was Sakurai’s “Advanced Quantum Mechanics” book.

When I was at UMIST in the mid 1990s we touched on qft in the 2nd year, and there was an optional particle physics course in the 3rd year.

Now that I thought about it more, I suppose such an âundergrad qftâ course could in principle cover quantum electrodynamics done using canonical quantization. What I had in mind was Sakuraiâs âAdvanced Quantum Mechanicsâ book.Thats what I had in my 4th year of undergrad. Renormalization was done using dimensional regularization. Almost no loops but we had some SM stuff.

Me,

We never got around to renormalization in senior year undergrad quantum mechanics. The furthest we got was Dirac’s equation and various solutions (ie. plane waves and hydrogen atom).

Amos,

On a very vague level, one can speculate that, since we use fermionic fields (valued in the Lie algebra of the gauge group) to deal handling gauge symmetry (a la BRST, or Fadeev-Popov), physical fermionic fields might also have an interpretation as a means for dealing with some symmetry. The natural symmetry to consider is diffeomorphism symmetry, or local translations. These fields though are spinor valued, which makes them rather different. I’ve played with various ideas for trying to do this, have some new ones I’m working on in the context of the BRST=Dirac cohomology framework. The fact that I still don’t completely understand this is one thing keeping me from finishing the BRST-Dirac paper.

JC,

My degree was a 5 year degree late previous century so this wasn’t even in my senior year. (though for some people it was).

We barely did renormalization but I sure remember dimensional regularization and people being baffled about integrating in d+e dimensions where d is an integer an e->0.

We did basic QED and some SM. Mainly calculating tree-level approximations to weak-force related events.

It wasn’t QFT as we later studied since things were motivated from Dirac’s equation and then upwards and not starting from a Lagrangian and QFT and deriving the rules. Things were very heuristic.

I didn’t like it very much but we certainly did some long calculations…

Does anybody have web references for the twistor reformulation of Yang-Mills in terms of contour integrals over Grassmannians?

Dave,

Look at

http://arxiv.org/abs/0912.0539 + earlier papers by same authors

and

http://arxiv.org/abs/0912.4912 + earlier papers by same authors