This week I’m getting ready for the start next week of the spring semester. I’ll be teaching the second half of our graduate course on Lie Groups and Representations, something I also did a few years ago, at which point I wrote up some notes and put them on-line. This year, since the students covered somewhat different material during the first semester, I’ll be covering some different topics, hoping to both write up some notes on the new topics, and improve the older notes. We’ll see how much of that I have time for. Throughout academia, others are also trying to figure out what they’ll be talking about during the new term, for example see Clifford Johnson’s recent posting. He’s teaching a course on string theory, something about which he seems to be a tad bit defensive. Actually his outline syllabus doesn’t really indicate what he will cover, referring to aspects of perturbative and non-perturbative string theory, gravity and quantum field theory, which pretty much includes most of modern physics. Perhaps, like some of the rest of us, he hasn’t quite yet decided what exactly to talk about…
A future course that some people might be interested in is a summer school to take place in Seattle on Lattice QCD and its Applications.
An American Physics Student in England has a review of QFT textbooks for beginners. He neglects to mention a couple of my favorites (maybe just because they are ones I learned from during my student days): Quantum Field Theory by Itzykson and Zuber, and Pierre Ramond’s Field Theory: A Modern Primer.
I saw the above link first at Dorigo Tommaso’s blog, which also contains all sorts of news about interesting results coming out of the Tevatron, including a new, more accurate value of the W-mass. See for instance here, here, here, and here. About the new W-mass measurement, there’s also a Fermilab press release, and an article in Nature. It may yet turn out that the Tevatron is the place where the Higgs is first seen.
Also in Nature is an interesting article by Frank Wilczek about recent lattice QCD results showing that QCD leads to a nucleon-nucleon potential with hard-core repulsion.
Notes from the talks at last week’s Gottingen Winterschule on Geometric Langlands are now available.
From Peter Teichner’s web-site, a new preprint by him, Hohnold and Stolz describing 8 different models for real K-theory, one of which is in terms of supersymmetric quantum mechanics. The paper is dedicated to the memory of Raoul Bott, whose periodicity theorem is a large part of this story.
From Michael Douglas’s web-site, there are slides from his recent colloquium talk here at Columbia on Supersymmetric Gauge Theory: an overview. He also has a new preprint out with Denef and Kachru entitled Physics of String Flux Compactifications. The autthors go over the arguments for the Landscape and devote significant space to discussing whether or not string theory is testable. They explain why hopes that one could use a statistical, anthropic argument to predict whether supersymmetry breaking happens at low or high scales haven’t worked out. There’s a somewhat mystifying claim that “in fact string/M-theory does predict a definite distribution of gauge theory and matter contents”, referring to various papers which don’t contain anything like a definite string/M-theory prediction of such a distribution.
As for the testability of string theory, the authors first note that while there are all sorts of exotic phenomena that one might imagine finding that are consistent with string theory, none of them are required by string theory, so:
Thus, while string theory can offer experimentalists many exciting possibilities, there is little in the way of guarantees, nor any clear way for such searches to falsify the theory.
They then go on to give what they see as four possibilities for testability:
1. “Swampland” arguments showing that string theory can’t possibly lead to a low energy effective theory that agrees with what we see. Unfortunately, there seems to be no such plausible argument, with all arguments of this kind so far only ruling out string theory as a source for very different physics than what we observe.
2. String theory must be true because there is no other possible theory of quantum gravity. They completely ignore LQG, but do admit that “one should not take this too seriously until it can be proven that alternatives do not exist”, mentioning the possibility of finiteness of N=8 supergravity.
3. Maybe the LHC will discover new physics that clearly is the result of a string theory compactification.
4. Maybe they will be able to make statistical predictions using the landscape.
These seem to me extremely weak and problematic arguments. 3 appears to be little more than wishful thinking that a miracle will happen and save the day, and all efforts over the last few years to pursue 4 seem to lead to insuperable difficulties for very fundamental reasons. In the end, the authors acknowledge this, writing “ultimately convincing evidence for string theory will have to come from observing some sort of exotic physics”, and putting their hopes in string cosmology, especially the hope of seeing networks of cosmic superstrings or signals in the CMB corresponding to non-linearities in the DBI action.
After this dismal summary of the situation and of prospects for the future, the authors decide to end with conclusions more or less directly opposite to the ones their arguments naturally lead to:
We conclude by noting that while the present situation is not very satisfactory, there is every reason to be optimistic… There are many well-motivated directions for improving the situation, and good reasons to believe that substantial progress will be made in the future.
Update: One more. There will be a public debate over the anthropic principle later this month, involving David Gross, Lenny Susskind, and others. More information here.
American Physics Student in England also forgot to mention Zinn-Justins’s Quantum Field Theory & Critical Phenomena”, which about as good a QFT textbook as currently exists. I always appreciated the fact that they explain Wilson’s perspective on effective field theory in the introduction, before doing even a gaussian integral.
Hi Peter,
Whats your favorite reference for Lie algebras? I remember you once mentioned Bourbaki.
Mark,
Kind of depends what you want to do with them. Bourbaki is unusually readable, but better might be a standard mathematical textbook, best-known algebraic one may be Humphreys. More for geometers, there’s Fulton and Harris. If you’re just learning the subject, a concise discussion is in the first part of the short book by Carter, Segal and McDonald. For physicists, who don’t care about arbitrary fields and really want to compute things, there are lots of references, and they might find Fulton and Harris readable. The beginning chapter on Lie algebras by Jurgen Fuchs in his book Affine Lie Algebras and Quantum Groups is nice, lots of computational details in Howard Georgi’s book.
I cut my teeth on Robert Hermann’s Lie Groups for Physicists (which I really cannot recommend) and Robert Gilmore’s Lie Groups, Lie Algebras and Some of their applications (now from Dover). Gilmore has a new version at:
http://www.physics.drexel.edu/~bob/LieGroups.html
Thanks, Peter, for making your notes available. I’ll read them soon. My major complaint about most books on Lie algebras is that they deal with matrix reps and ignore reps via differential operators as is done is courses in differential geometry.
In line with the variety of items above, I thought I would pass along a blog posting about a new open access journal, mentioned in a comment on CV:
PhysMath Central
(..open access, but commercial.)
As a student, I preferred Ramond over Itzykson and Zuber because it wasn’t so prohibitively big. IZ is a better reference, though.
A second drawback of IZ, which I have become increasingly aware of during recent years, is that small print is unfriendly to middle-aged eyes.
The book by Sattinger and Weaver on Lie groups and algebras is not
bad. It’s pretty rigorous, and much of it is in language physicists can
understand. The best thing about it is that it’s short. The chapters on classification of semi-simple Lie algebras are pretty good (except the proof of Engel’s theorem is impenetrable – I had to find that someplace else).
I also second Peter W.’s endorsement of Segal’s lectures, in the book
with Carter and MacDonald.
A minor quibble with Wilczek’s nice article: When he mentions the limitations of the calculation of Aoki & co (e.g. quark masses larger than their physical values) he neglects to mention that the calculation is also “quenched”, i.e. fermion determinant in the path integral is set equal to 1 and hence the effects of vacuum polarisation by quark-antiquark pairs are ignored. I’ve no idea to what extent this affects the result; presumably its qualitative features – in particular the hard core repulsion – would remain, but quantitatively the inter-nucleon potential could be quite different. At any rate, going from quenched to full (unquenched) QCD certainly affects the results for hadron masses, reducing the discrepancies with experiment from ~10-30% to ~1-2% in the high-precision lattice calculations that Wilczek mentions (but surprisingly doesn’t give a reference to). The problem with having to do calculations at larger quark masses is not so serious since one can extrapolate to physical values (via chiral perturbation theory), but there is no similar solution for unquenching – you just have to get a powerful enough computer and do unquenched calculations from the beginning. So this seems a more serious challenge as far as removing the limitations of Aoki & co’s work goes.
Btw Peter, do you ever think about returning to the lattice? There are interesting things going on these days, including mathematical things. E.g. it is now understood how to formulate fermions on the lattice with chiral symmetry, exact chiral zero-modes etc, so the notion of index makes sense and one can ask whether it coincides with the (your!) lattice topological charge of the background gauge field, as it should (for sufficiently smooth fields) according to index theorem. Numerical studies indicate that this is the case, but there is no analytical proof so far. So if you ever feel like returning to this topic there’s a research problem for you right there.
Jacobson’s old book on Lie algebras is quite readable and very cheap (its a Dover). It also goes into more depth than most the current standards (it’s an algebra textbook not a geometry textbook).
amused,
I actually got interest in the things I’ve been working on for many years when I started thinking about chiral lattice fermions. It has always seemed to me that which lattice gauge theory does a beautiful job of taking advantage of the geometric nature of gauge fields, the discretization of fermions completely ignores the geometry of spinors, and maybe there is a better way of doing things.
Anyway, this led me to other questions and problems I’m still struggling with, going back to the lattice problem to see if anything I’ve learned about spinor geometry can be useful there is on my ToDo list, something I should try and find time to work on.
Off topic, but talking about books, does anyone know a good introduction to differential geometry? Last time I looked at the first chapter of Do Carmeno, it seemed to have bunch of errors. Spivak is very nice, but perhaps a bit thick. Any other suggestions?
Peter,
Sorry if this is getting off-topic, but chiral fermions on the lattice are in pretty good shape these days in the overlap/Ginsparg-Wilson formulation. It certainly doesn’t have the obvious beauty and naturalness of the lattice gauge field formulation, but on the other hand, given the obsticle of the Nielsen-Ninomiya no-go theroem, it might be as good as it gets. At any rate, many things work out as they should, including chiral gauge anomalies and their relation to families index theory. People haven’t tried to describe general aspects of spinor geometry in this formulation though – it’s all been usual spinors coupled to gauge fields on flat spacetime so far.
Peter,
As amused said, the problem is solved for QCD. The status of
spectra and some other numbers of QCD is actually looking
very good.
On the other hand, regularizing the standard model with the
lattice is in poor shape. Genuine chiral fermions are a problem.
The Nielsen-Ninomya theorem is a potential problem with
any cut-off, not just the lattice (in fact, Nielsen and Ninomiya
did not use a lattice). It may be that this problem points
to some new physics beyond the standard model.
Peter (Orland),
Can you elaborate on that? I was under the impression that Nielsen-Ninomiya comes from some index theorem based on putting the theory on a torus. So it looks to me like it has more to do with IR regulation than with UV regulation. So couldn’t you just take the finite-volume regulator to be on some other topology with a different value of the characteristic class appearing in the index theorem, and take the infinite volume limit of that?
(I realize I might have some major misunderstanding here, since I’ve only heard the Nielsen-Ninomiya theorem discussed briefly a couple of times.)
Neither American Student in England, nor any of the respondees has mentioned the new player in town, by Mark Srednicki, chair of the physics dept. at UCSB: http://www.physics.ucsb.edu/~mark/qft.html
In my humble opinion, it will shortly blow Peskin & Schroeder Out of the proverbial water. Our QFT instructor, immediately upon discovering Srednicki’s new text, not even due out until Feb., dropped P&S immediately in favor of it. Check it out !
Peter Orland,
It is true that the lattice formulation of the Standard Model hasn’t been accomplished yet, but nevertheless I still consider the lattice formulation of chiral gauge theories (not just vector ones like QCD) to be in relatively good shape. The fundamental obsticle represented by the N.-N. no-go theorem has been overcome by the overlap/Ginsparg-Wilson approach, and it has been shown to correctly reproduce the anomaly structure of the continuum formulation. What remains is to construct the chiral fermion measure (or “perfect phase” in the overlap terminology) such that the lattice theory is gauge invariant when the usual (continuum) anomaly cancellation conditions are satisfied. This has already been done for gauge group U(1) but not yet for the nonabelian cases (including Standard Model). There are no fundamental obsticles to accomplishing this; it is “just” a hard technical problem. The prospects of this being sorted out any time soon are bleak though since no one seems to be working on it. (Some of us would like to but then we’ld soon be out of a job…)
Amused,
I am not so sure that the non-Abelian case is just a technical problem.
Also the U(1) solution is very complicated, and it is not so clear that
it is satisfactory.
I-Z was my textbook, and I do not like it. Actually I wonder if someone learning mainly from it got to do research on QFT. It was a transition book, for a generation who was too late to use the Bjorken et al books and too early to enjoy the new books.
Re: Srednicki, I too thought it was better than average, and it is nice to find a QFT reference that is more the thoughtful, useful pedagogical text and less of the cookbook.
Having said that, I did feel the need to send an e-mail to the author pointing out a problem with the common SL(2,C) conventions used in the supersymmetry community. As in some other areas, I seem to be the only one who worries about these things, but this particular problem cost me at least a day of my not-very-valuable time when I was a graduate student.
Peter O.,
“I am not so sure that the non-Abelian case is just a technical problem.”
All I can say is that it’s my strong impression after having worked on it and tallked to others who have worked on it.
“Also the U(1) solution is very complicated, and it is not so clear that it is satisfactory”
If you have found something unsatisfactory in Martin Luscher’s argument I suggest you write and tell him about it. (It seemed fine to me when I studied it.)
amused,
Just so I understand what you are saying….
Are you stating that it has been resolved that chiral fermions, e.g. neutrinos have a good lattice formulation?
… and that everything works when coupling said particles to gauge fields?
I am looking over Luescher’s papers to say exactly what he says.
Peter O.,
Before coupling to gauge fields, chiral fermions such as neutrinos do indeed have a good lattice formulation now. And they continue to do so if you couple to U(1) gauge fields in an anomaly-free representation. For coupling to nonabelian gauge fields the situation is still unresolved at the nonperturbative level. (At the perturbative level things have in fact been resolved and everthing is fine.) As I mentioned above, the remaining problem in this case is to construct a chiral fermion measure such that the theory is gauge-invariant when the fermions live in a representation of the gauge group which satisfies the usual (continuum) anomaly cancellation conditions. (There is a priori an arbitriness in the phase of the chiral fermion measure, and the challenge is to determine a suitable phase factor (as a function of the lattice gauge field) such that the chiral fermion determinant becomes gauge invariant.) Actually there has already been substantial progress on this for electroweak gauge group U(1)xSU(2) but not yet a complete solution. It’s a difficult technical problem – as you mentioned, the U(1) case itself is already pretty complicated – but i don’t see any fundamental obsticle to things working out.
This is getting quite off-topic from Peter W.’s original post, I hope we aren’t testing his patience too much… Anyway I have to go to bed now (it’s 4am here) so won’t be able to get back to this until tomorrow…
Dr Woit, you may enjoy Sharon Begley’s column in the WSJ January 5: “Giant Swiss Collider May Reveal Secrets About Origins of Mass”
“If the LHC creates superpartners, the results will be spun as fiercely as a political campaign debate. String theory, which asserts that the basic constituents of matter are tiny vibrating strings that exist in 11 or 12 dimensions, requires supersymnetry. Thus stringsters may hail signs of superpartners as their long-sought vindication. But rival theories posit superpartners too, so if the LHC finds them it wouldn’t uniquely support string theory.”
“Supersymnetry is a vital part of string theory, so if the LHC doesn’t find it, that would argue strongly against string theory,” says physicist Lawrence Krauss of Case Western Reserve University, Cleveland. “If it is observed, you can say that string theory has not been disproved, but not that it has been validated.”
The article also contains quotes by Lee Smolin. I can post the whole thing if you wish.
More or less off topic – Anjana Ahuja (the London Times 15th Jan.) flags up NEW (and Lubos’ Reference Frame) as an ‘exciting arena for gladitorial spats … over string theory’. Somehow, she manages to make the Sage of Harvard appear to be the more reasonable of the combatants: his mildest ever invective (‘crackpot’) is paired with what must ahve been Peter Woit at the end of a very hard day.
Chris Oakley: thanks for posting a link to your explanation. I will have to think about this some more. Do you have a link or reference for Penrose’s conventions? I don’t have his book handy. I want to make sure that my conventions are consistent.
If we take the contravariant symbol to be the inverse of the covariant symbol (a la the metric in GR), which would make them negatives of each other when viewed as matrices, we have eps^(ab)*eps_(bc) = delta^a_b. Then we have eps^(ac)*eps_(cd)*eps^(bd)=delta^a_d*eps^(bd)=eps^(ba)=-eps^(ab). Thus it doesn’t seem that having them be inverses of each other and obtaining each other by raising and lowering indices are consistent, assuming raising and lowering are done with the second index. I hope my notation is clear. contravariant indices are preceded by a ^, and covariant indices by a _. I think if you had them not be inverses of each other, but be the same then you would get eps^(ab) by raising the indices of eps_(ab). Is that what Penrose does?
For those wondering what “lost soul” is referring to, see
http://www.timesonline.co.uk/article/0,,20909-2547760,00.html
The quote is from here
http://www.math.columbia.edu/~woit/wordpress/?p=490
and taken out of context. It was part of a response to commenters who wanted me to remove links to Lubos’s blog because of the racist nonsense he spews.
I think that String Theorists should be grateful to Lubos for his efforts in keeping public interest alive in a subject that might otherwise die for lack of experimental support. It would appear that many of them are.
King Ray: I will try and keep this short, but Penrose’s handling of SL(2,C) – which is of course a pre-requisite to twistors – I got from lecture notes, not a text book, although I am sure that there will be printed references somewhere. The notes on my web site spell out my understanding of this matter. The covariant-contravariant relationship does not require the ε identities as you give them – it only requires that the transformations under group action are inverses of each other. Your identities in fact almost apply with Penrose’s conventions, apart from a few sign differences. Yes, obtaining ε^{AB} by raising the indices of ε_{AB} is the thing that matters, and it is indeed what Penrose’s conventions guarantee. My point is that if one does not have the same rule for raising/lowering indices of preserved tensors as other tensors, then all hell will break loose.
Chris, if the covariant and contravariant 2D Levi-Civita symbols are not inverses of each other, and you use the second index of each to raise or lower indices, then if you raise a spinor’s index and then lower it you get back the same spinor. If your LC symbols are not inverses of each other (i.e., are the same), then you get the negative of the spinor you started with (like a complex structure squared acting on the spinor since eps^2=-I). I think the way others are doing it is ok so long as you know that when you raise the indices of the covariant LC symbol you get the opposite of the contravariant LC symbol. Maybe they should have two different symbols. Maybe all could be made good if you raise with one index and lower with the other and make covariant and contravariant LC symbols equal? I’ll have to see what Penrose does.
I looked up Penrose’s raising and lowering conventions in Penrose and Rindler, Spinors and space-time, Vol. I, and Penrose has covariant and contravariant LC symbols the same, and lowers with the 1st index and raises with the 2nd. I guess a good mnemonic would be L1R2.
Chris, thanks for bringing up this issue. I may adopt Penrose’s convention. The only problem I have with it is remembering which issue to raise with and which to lower with, which is why I always preferred to raise and lower with the 2nd index.
In summary, I think Srednicki’s conventions are consistent, you just have to keep in mind that eps contravariant is not gotten from raising the indices of eps covariant. I can’t see any problems if you keep that straight.
It must be 20+ years since I looked at Penrose’s book, so thanks for getting me to do that. I also saw a nice set of identities in there that are all worked out. Penrose and Rindler also have a nice appendix on spinors in n dimensions in Vol. II.
King Ray – right. But one must have ε_{AB} = ε^{AB} or the bad things alluded to in my note will happen. See also here, eqs. 3.31-3.35.
Yes, but that means that ε_{AB} is not a tensor. I would definitely call that a problem. It means that every time you raise or lower its indices you have to remember to change the sign as well.
PS: If you are who I think you are (I have only just twigged), then I apologise for a somewhat brusque reply to an earlier post from me. Peter, fortunately, deleted it.
Chris,
Thanks for the link to your document, I will look at it more carefully later.
I agree with you 100% in that it is an abuse of notation to use epsilon for both contravariant and covariant LC symbols if you use Srednicki-like raising and lowering. Like I said above, they really should have different symbols in that case, but using the same symbol is ok if you don’t assume that they are related by raising and lowering indices. Penrose’s convention is definitely less confusing as far as that goes, although harder to remember since you raise and lower with different indices.
I did not see the post you referred to, so no harm was done. I am not sure who you think I am, but I am probably not that person unless Peter told you who I am. I prefer to be anonymous, one reason for which is that I have many friends and acquaintances in the string community. In the past, I have gone on long walks with Penrose and Newman on a number of occasions.
Also, yes, you must have eps^(AB) = eps_(AB) in Penrose’s convention as I alluded to previously. Otherwise raising and then lowering gives you a minus sign as I stated above.
Peter,
I don’t follow your blog, so there’s no way that I know whether you know or you don’t know about the following workshop in KITP. with both string theorists and numerical GR and looooooooooooooooooooooooooooopists.
Here is the link:
http://online.kitp.ucsb.edu/online/singular_m07/