I really am trying to ignore Lubos, but there’s just too much material…

Back in early 2004, after it became clear that Cambridge University Press was very unlikely to ever publish Not Even Wrong due to intense opposition from string theorists, I tried sending the manuscript (together with the Cambridge referee reports) around to a few other university presses to see if any of them would be willing to publish it. The response I got from two editors at well-known presses was positive comments about the content of the manuscript, but:

*I think it’s too controversial for a university press to publish.*

from one, and from another

*it is extremely unlikely that a proposal as controversial as yours would be accepted by the [governing board].*

This made clear exactly how much of a “free marketplace of ideas” exists for debate about string theory within this part of the publishing world.

An editor at Princeton University Press wrote back after considering the manuscript for a week or two with a form-letter rejection informing me that “we must often forego formal review of promising manuscripts or proposals such as yours”. I assume that, as I expected, the editor had discussed the manuscript with one of the local string theorists and thus been convinced not to pursue it.

With Roger Penrose’s help, finally late in 2004 the British publisher Jonathan Cape bought the book, planning to publish it in Britain and sell the U.S. rights to an American publisher. During the first part of 2005 I worked a bit more on the book and it was copy-edited, and by the early fall the people at Cape were in negotiations with various possible US publishers, negotiations that I had little to do with. In November the editor at Cape told me that Princeton University Press had rejected the book as “too controversial”. The next month US rights were sold to Basic Books.

I had no idea about this at the time, but it seems that someone had advised Princeton that the appropriate person to review this kind of manuscript and give an unbiased opinion about it was a Harvard string theorist with a well-known blog named Lubos Motl. Lubos has now posted his report, together with the proud claim that “a serious publisher whose name was edited used [it] to scrap the project.” He cleverly hides the true name of the publisher in question as “P. University Press”.

The report makes clear what Lubos was going on about in some of the incomprehensible parts of his Amazon review. I responded to that review here, but couldn’t even figure out a lot of what he was talking about there. With his detailed report with page numbers, this is now clear.

He was definitely on his best behavior. The report is not obviously a rant, and even includes some positive comments. He carefully went through the manuscript making many sorts of copy-editing suggestions (e.g. changing English spellings to American) and suggested a large number of rewordings of the manuscript that would make what it said agree with his vision of reality (but not mine).

Anyone interested can go through the report, compare it to the book and judge for themselves whether Lubos’s extensive criticisms make much sense. Responding to his 17 pages filled with misinterpretations of what I wrote and tendentious claims about string theory is something I don’t have the time or energy for, but I’ll respond to his summary where he says that the book should be rejected because of its “many serious and elementary errors.” He lists these as:

1. I don’t know the difference between a GeV and a TeV. This is based on one typo, on page 32, where, after writing that the center of mass energy is at the LHC is 14 TeV, I mention that it might be possible to double this energy by doubling the strength of these magnets, and “28 GeV” is an obvious typo for “28 TeV”. This typo is fixed in the US edition, thanks to the fact that he makes this argument against the book in his Amazon review.

2. He objects to my pointing out (page 179) that in a theory with broken supersymmetry the vacuum energy scale is too large by a factor of 10^{56}, wanting me instead to say that supersymmetry “improves” the vacuum energy problem with respect to non-supersymmetric theories by a similar size factor. What I wrote is correct.

3. On page 35 I mention that the neutrinos produced by a muon collider interact weakly, so will go through the earth and produce a radiation hazard when they emerge many miles away. Lubos claims that this is wrong, that “neutrinos with hundreds of GeV of energy interact strongly”. This is nonsense. What he has in mind though is not really a “strong” interaction strength, but an electromagnetic interaction strength. He’s right that at hundreds of GeV (way above the W and Z masses), there is electroweak unification, and the weak interaction and electromagnetic interaction strengths are similar. However, he seems to be making an elementary mistake: the neutrinos involved will be hitting a fixed target, so the energies involved will be much lower.

4. He repeats a mistaken comment that I once made on my blog about about SU(2) and SO(4), one that has nothing to do with what I write in the book. His excuse for introducing this is that on page 49 I refer to “axes of rotation” in 4 dimensions, complaining that I should have explained that in 4 dimensions rotations are specified by choosing not a one-dimensional axis, but a two-dimensional plane. It’s quite true that I was simplifying things, not explaining that in N dimensions an “axis of rotation” is N-2 dimensional. Explaining that more carefully was not something I wanted to get into. Perhaps he’s right that it would be better if I put “axes” here in quotes to keep people from making the wrong assumption that he’s making.

5. He finds something wrong with the fact that even though I explicitly say that the physical Hilbert space is the trivial representation of the gauge group, I speculate that understanding the non-trivial representations of gauge groups is an unsolved mathematical problem whose solution might tell us something interesting about gauge theory. This is clearly labeled as speculation and perfectly accurate as written.

Anyway, now I know why Princeton rejected the book, although I still have no idea who put them up to choosing Lubos as a referee.

For more about Lubos and the controversy over string theory, there’s an article in the Frankfurter Allgemeine Zeitung (in German). Lubos comments that “virtually all well-known theoretical physicists” think as he does, but that only he (together with Susskind) is willing to fight compromise with very stupid people and crackpots like me. He warns “to the polite big shots: the more silent you will be the more loud the blunt opinionmakers such as Susskind or your humble correspondent will have to be.”

Anyway, now I know why Princeton rejected the book, although I still have no idea who put them up to choosing Lubos as a referee.I find this puzzling too, as sending

Not Even Wrongto Lubos is a bit like sendingThe Da Vinci Codeto Torquemada.At the end of the day, though, publishing is a

business… even without any referee reports it was obvious that you had the credentials to write such a book, and the question really was, would people buy it? And asThe Da Vinci Codeproved, even things that are arrant nonsense can sell if they are controversial.Just a brief comment about particle beams AFTER they go through particle detectors ~ “radiation hazard”. (One can have in mind neutrino beams.) The beamlines are typically angled to that the beams go flying up in the air and enter the atmosphere and outer space. This is certainly the case at Fermilab, where the beamlines are angled upwards. By the time the neutrinos etc clear the FNAL boundary, they are in the air above any homes outside FNAL. (No skyscrapers there.) No radiation hazard. I believe there is a no-fly zone where airplanes stacked for O’Hare do not fly, to avoid their electronic instrumentation being hit by the beams. I have no doubt the ILC or a muon collider will be designed the same way. Someone from FNAL can comment in more detail.

Chris,

University presses are funny businesses, not completely driven by the profit motive. It is generally part of their mission to have high intellectual standards and sometimes publish things that might not make much sense financially. In the case of my book, the controversial nature would suggest that it was likely to sell better than much of what they publish and might make them some money. The argument against publishing it would be that it didn’t meet their intellectual standards, and this is what a reviewer is supposed to provide an evaluation for.

Geez. My review is ony 11 pages right now….

(Which I will finish up eventually, but right now I’m focussing most of my time on Hochschild cohomology.)

well, at least now you can be sure that lubos wasn’t lying when he said that he has read your book.

Lubos Woit,

It is dishonest to comment on a MANUSCRIPT and claim that it is a review of a BOOK.

Anyway, now I know why Princeton rejected the book, although I still have no idea who put them up to choosing Lubos as a referee.Maybe they support the book, and were cleverly using their position to garner great publicity for your book via Lubos!

Peter,

I have to say that, while I do think it is a proven fact that the trouble your manuscript faced when trying to become a book is almost entirely due to the string mafia, I also believe that the point you seem to make at the start of the post above is prone to criticism: you seem to imply that when somebody says “this book is too controversial” (s)he means “this book goes against the mainstream thinking of the big mushrooms and it would bother them”.

I think there are two separate issues. We agree on the mafia, but I think (some) University Press editors might be honest when they say they prefer to avoid publishing divulgation material which is so focused on criticizing a theory en vogue.

The fact that there was quite a bit going on under the tables during the review process of course might make the above detail irrelevant, given the direct attempts at having the book rejected. But in principle, UPs might be right if they think scientific controversy is better solved at the blackboard than in a media fight… Indeed, controversy usually sells well, as somebody pointed out above – but UPs are right in ignoring that. You said it – there aren’t just business issues there.

That, of course, does not mean the book wasn’t needed or useful… Quite the opposite. Thank you for writing it!

T.

Form Lubos review:

I find it completely necessary to mention the names of Andrew Strominger and Cumrun Vafa – the authors of the pioneering work that has shown that string theory gives the right value for the black hole entropy.What was actually discussed by Strominger and Vafa:

The Bekenstein-Hawking area-entropy relation $S_{BH}=A/4$ is derived for a class of five-dimensional extremal black holes in string theory by counting the degeneracy of BPS soliton bound states.http://arxiv.org/abs/hep-th/9601029

This is definitely not the same as what is suggested by Lubos — that the result applied to all black holes (whether charged, uncharged, or rotating) — which we know was the original Hawking result. In fact, read the Strominger and Vafa paper, look at the result, and you see it does not apply for uncharged (electric charge) black holes…It also relies on the conjectured existence of Axions and that the black hole has large Axion charge.

Sadly, the same tremendously overstated misrepresentation appears on Vafa’s own web page:

http://www.physics.harvard.edu/people/facpages/vafa.html

Among these one can name the discovery of Strominger and Vafa that the Bekenstein-Hawking entropy of a black hole can be accounted for by solitonic states of string theory and also the relation between geometry and field theories that arise through string dualities, a topic known as “geometric engineering of quantum field theories”.The Lubos review is neither objective or balanced…but that is not new for him.

What is disturbing is the thought that Princeton may have actually relied upon his review.

But don’t worry, DOVER will pick up N.E.W. and re-print is as a classic đ

Tommaso,

You may be right that part of the concern with “too controversial” is a fear of more heat than light being shed. In any case it’s hard to know really what is behind editor’s decisions. They’re often not going to give truly honest reasons in a rejection letter, for lots of reasons, including simple politeness.

In this case though, from going through the process with the editor at Cambridge, I think I can see the problem that university press editors faced when considering the book. The book makes strong claims that some very accomplished (as well as powerful) people are wrong about something, and the issues involved are both highly technical and understood by relatively few people. The editors involved don’t have the background to be able to decide for themselves how good my arguments are. When they ask experts, they find non-string theorists willing to support what I have to say, and string theorists often vehemently claiming I don’t know what I’m talking about.

They end up in a very uncomfortable position of having to decide whether to put some of their reputation and the reputation of their institution behind me, without being able to be sure whether I’m right. In addition, these presses often have some faculty body that has to sign off on publication decisions, so the editor has to worry about whether he or she can convince them also. Not an easy decision to make.

So, I’m not too surprised that these university presses didn’t want to get involved. I am surprised that Princeton University Press did decide to get involved to the extent of commissioning a review, then chose someone obviously inappropriate to do this.

Peter,

No doubt there are many of LM’s comments that are trivial or even silly, but a few have apparent substance. I’m thinking of the critique of your statement on p 159 of the manuscript “string theory is not a background independent theory.” I looked at the relevant chapter of Polchinski, and he does indeed claim that ST is not restricted to just one background. Could you clarify this issue?

Do you plan respond to his more substantive points?

CIP,

I hadn’t planned on going through Lubos’s whole document and responding to all it, I just don’t have the time, and mostly I don’t think it would be very enlightening.

As for the “background-dependent” business, you could consult extensive battles over this between string theory and LQG proponents on this blog and elsewhere (don’t even think of starting this up again Who or Aaron…).

The fact of the matter is that to even write down at all what you mean by a string theory, you have to first choose a background. Different backgrounds give different physics. This is what I mean by “background-dependent”. What I wrote is completely accurate. String theorists have an argument, that infinitesimal deformations of the background space still leave you in the same theory, just a different state.

One could go on and on about this issue, it quickly gets confusing because string theorists love to mix up what is actually known and corresponds to something they actually can understand and write down, and what they don’t understand, but hope to be true, with various degrees of evidence for the hope.

I really, really, really don’t want to try and moderate the N’th unenlightening discussion of this issue here. If people absolutely can’t restrain themselves, go ahead and provide a link to a discussion of this somewhere else that you agree with. Just don’t try and fight out the issue here.

Hey, I saw it in Amazon.com that Lee Smolin is coming out with a new book in September. Curiously it is about String theory as well. Not to worry, I will buy both.

Hi Peter,

I read Lubos review. I found it very nice actually. If I think of some reviews I receive on my papers, this is a very well founded, reasonable, though biased list. I can’t avoid having the impression that the publisher was only looking for a confirmation that the book would be too controversial, otherwise the mentioned points could easily have been revised.

@Energex,

Lee’s book is not about string theory. It’s about the trouble with physics. Reading Peter’s last post, it seems that’s gotten more of a trouble with physicists lately.

Best, B.

Dear Peter,

You’re saying that:

“different backgrounds give different physics. This is what I mean by âbackground-dependentâ.Could you please make more precise the definition of the term “different backgrounds”?

(1) Are you claiming that the background g’_{ij} = exp(2)g_{ij}, for example, will give different physics from the background g_{ij} ?

(2) And are you claiming that the sum over topologies (of the world-sheet) is not determining the interactions (of the superstring)?

(3) And – in relation to (1) and (2) – in analogy that only “background independent” formulations of, for example, QCD are relevant?

(4) And that

string field theoryis not a background independent formulation of string theory (in the sense to be defined above)?Cheers, Kasper

Kasper,

1. No

2. That has nothing to do with this question.

3. That has nothing to do with this question.

4. Yes

Is it possible that Lubos’ role in the review process was the result of a misunderstood joke?

Publisher A (doesn’t read blogs): Who should we get to review this?

Publisher B (reads blogs): (sarcasm) Of course, we should get Lubos Motl from Harvard.

Publisher A, impervious to sarcasm, sends manuscript the next day.

About neutrinos: they produce radiation hazards in TeV-energy muon colliders because 1) muons decay to neutrinos; 2) TeV neutrinos interact with matter LESS strongly that electrons, photons etc, so that building the collider underground is not enough to shield neutrinos: they exit making occasional showers.

PS: how many books had a Harvard professor as spell checker?

Dear Peter,

I forgot to make precise that the first question is not (1), but (0), or:

(0) Could you please make more precise the definition of the term âdifferent backgroundsâ?

and I don’t hope you’ll answer

0(1). No,

or

0(2). That has nothing to do with this question.

In saying that background A is different from background B, one must have a way of classifying different – and consequently equivalent – backgrounds, or – loosely speaking – a metric topology on the space of backgrounds. For example, I would use a metric where background

A’ = exp(2) A

is at zero distance from background A.

cheers, Kasper

Hi Peter,

Peter, with a heavy heart I must state my dismay and disappointment with your actions – in regards to your various opponents, so far.

Lubos – with all his grandiose utterances – is nothing more than a rambler. (His own utterances will testify amply to that.)

When will you – if ever(??) – start discussing physics again?

If, at any time you do, please follow on:

I have been trying to find some useful information on the topics we discussed upon earlier:

1) Dirac operators, and

2) Gauge fields.

Landmark texts / works?

Do you by chance happen to know the standard references to consult? (Textbooks, Arxiv, etc.)

I am also trying to purchase the text by Jouko Mickelsson: Current Algebras and Groups.

Finally, do you happen to know what the following texts are primarily studied for:

1) Gauge Fields: An Introduction to Quantum Theory – L. D. Faddeev, A. A. Slavnov;

2) Gauge Fields and Strings â A. M. Polyakov;

3) Quantum Field Theory: From Operators to Path Integrals â Kerson Huang;

4) Operator Methods in Quantum Mechanics â O. L. de Lange, R. E. Rabb;

Thanks,

Stefan

LM writes on his blog: “People, including those with degrees, tend to trust the media, and because the media have been producing crap about physics most of the last few years, it more or less means that most people who rely on media inevitably believe this crap.”

You don’t think Lubos means the media-hype around string theory, do you?

I’m wondering whether delaying Peter’s book has increased its impact, because as each year goes by, the claim of progress becomes less plausible.

Stefan,

When did you ever get the impression that Peter knows anything about cutting-edge physics? He never did and doesn’t now. Thats why he has this blog–the only way to make a name for himself, which is much easier than actually doing research. He keeps promising to talk about physics, but never does, because he has nothing interesting or substansive to say. You should check out his (in)famous ideas on the embedding of SU(2) into SO(4) to see the calibre of great ideas he thinks up. This isn’t meant to be insulting to Peter, only explanatory. He has much more fun knocking down people who are seriosuly working for years on interesting difficult ideas, especially when it requires no intellectual effort of him. Thats why almost no real theorists–string theorists or not–respects or even pays any attention to him. If you’re interested in substansive physics discussions, ignore blogs and the mediocrities and crackpots they attract altogether. Peter has been promising to tell us about his fantastic ideas for research for a long time now, and I predict it will never happen.

Kasper,

I have no idea what the precise set of consistent string theory backgrounds is or what the correct metric on it is, and I don’t think anyone else does either. But as far as anyone can tell, there are many of them, they give different physics, and you have to choose one before you can even write down what your theory is and try and see if you can extract any physics from it.

Stefan,

There are by now hundreds of places you can read the basics of gauge theory and find out what the Dirac operator is. Some of the books you mention contain this, there are many others. You first need to understand quantum Yang-Mills theory at the level of the books you mention. Another one that is highly readable is Pierre Ramond’s QFT book.

Once you understand that, then I’d suggest learning about the mathematical context of these things. I keep encouraging people to read Atiyah’s expository articles. The Mickelsson book you mention is much more advanced than the other books, but also highly worth reading, it’s a much more research level document.

AC,

You know, you anonymous pluckers really should try and find at least one more mistake I’ve made in what I’ve written here and in the book. I’m sure among the by now thousands of pages of stuff there’s at least one more.

Peter, keep up the good work. I think your criticism of string theory is contributing more and doing a greater service to physics than all the string theorists combined. They have no aesthetic compass.

Peter,

I have now finished reading the UK version of NEW. As someone who works in the molecular/chemical end of science (among other things) and not in HEP, I found the book difficult in spots. However, by the end I thought you had given a clear overview of your position on the current state of theory in HEP. The essential arguments are there. If the string folks want to say that’s not correct and explain why, then that should help understanding in the field. WRT to Lubos, I don’t think you should expect much. Based on my interactions with him, I find that he is not interested in the reasoned discussion of scientific issues. He’s only interested in telling people that his view is correct and everyone else is wrong because they’re too stupid to see that he’s always right. A particular thing that I find annoying is his over and over misquoting someone and concluding the person is stupid based on his misquote. In summary, I think NEW is thought provoking and thus adds something positive to HEP discourse. I think a full and frank discussion of these issues will make HEP a stronger field. I hope that happens. As you mentioned, last week, if some of the braneworld models that may be testable at the LHC are supported, you’d modify your views. That’s what a scientist does. Hopefully other people will take note, and adopt a similar attitude. NEW, the book, is, I think, a very positive contribution.

Best,

David

I do not usually reply anonymous postings but will reply this one.

Anonymous Coward said,

Right! I do not remember Peter Woit (not other here) claiming that. I remember string theorists claiming that they are making the TOE (i.e. claiming that they know anything about anything).

Right again! This is reason that string theorists wrote so many popular books, give talks for outsiders, submit articles to generic magazines, give interviews to media. Unfortuntately, after 40 years there is none serious work about string theory, none prediction, none explanation of misteria of SM and GRâŠ

Would i remember you how many wrong stuff is said each day by string theorists? None serious physicist working in quantum measurement takes Brian Greene ideas about the topic seriously (Dyson was very clear about this a few years ago). There is moreâŠ

Do you mean that serious people who claim that string theory predicts gravity (Witten), that string theory is the language God wrote the world (Motl), that the CC is explained by the antrophic principle over a quasi-infinite Landscape (Susskind)?

Hum, when you ask Witten how string theory predicts gravity he replies that really is a postdiction, when you ask Motl about derivation of some known stuff in physics he replies insulting (as usual in him because string theory predicts/explains nothing), when you follow details of the Susskind approach you coincide with Gell-Mann: absurd.

I know that string theorists -and others- usually do claims without basis. Could you provide any reference or statistical data for your claim please?

Then why are you writting here?

I do not know, nobody knows. However, is not that the history of last 40 years of research in string theory? During 40 years we receive claoims about all kind of fantastci stuff could be done from string theory and how M-theory would revolutionate our views about space and time? result?

1) String theory is unable to explain data can be explained with common theories as SM and GR. String theory is compatible with nothing of this world.

2) Nobody knows M-theory is (if it really exists).

3) Since string theory cannot explain stuff as CC using scientific method the scientific method is abandoned by metaphysical cuasi-religious stuff (instead abandoning string theory nonsense).

Juan R.

Center for CANONICAL |SCIENCE)

08 03 06

Hello Peter:

Congratulations on your book reviews etc. And I am curious about what you or anyone else thinks about Mr. Thiemann’s newest paper on AQG? Pls pardon me if you have already written about this. I asked Lubos, but to no avail. I don’t think my knowledge is developed enough to analyze the paper thoroughly, perhaps you or someone else can? Thanks.

Mahndisa,

I took a quick look at the papers, they look interesting, but to really understand them would take a lot more time, since I’m no expert in this area. I hope to find the time someday to read them more carefully.

08 04 06

Thanks for the response Peter. I have a feeling, a thorough analysis of that set of papers will take a lot of time yet;)

Dear Peter

Now, if what you are saying, that

I have no idea what the precise set of consistent string theory backgrounds is or what the correct metric on it is, and I donât think anyone else does either.is true, then

(1): how can you claim, that

there are many of them, they give different physics????

and related to my question above, which you considered irrelevant to the discussion,

(2): if your statement, that

you have to choose one before you can even write down what your theory is and try and see if you can extract any physics from it.is true, then are you applying the same critisism to theories like

QCD, orQED????

Cheers, Kasper

Kasper,

This is about Lubos’s claim that I’m wrong to say that string theory depends on a choice of background, this has nothing to do with QCD and QED. You’re trying to start up a different argument that has already taken place here many times.

Your attitude seems to be that string theory is background independent since you don’t know what the possible backgrounds are or how to calculate anything in them. I suppose that’s a consistent point of view…

Dear Peter,

I think my question (2) was relevant for the reason implied above, but of course it is up to you if you want to answer the question (2), or not.

And whether Lubos claims this or that has nothing to do with my question – actually,

I’m just trying to follow your line of thinking;and I think it would be relevant to get an answer to my question above:

(1): how can you claim, that

there are many of them, they give different physics??

But of course, I haven’t read your book yet…

best, Kasper

This comment is too long and has too much math in it, but I thought readers might be interested in an elaboration of Peter’s “wrong” ideas about subgroups of SO(4). Disclaimer: I am not a working mathematical physicist, just a hobbyist with an unfashionable interest in physics on 4-dimensional manifolds of Euclidean signature and complicated topology.

If I’ve read the infamous “SU(2) embedded in SO(4)” error correctly (“Wick Rotation”, Feb 28, 2005), it’s not even (all that) wrong. Yes, if you insist on viewing SU(2) as an _invariant_ (normal) Lie subgroup of SO(4) and imposing a _group_ structure on SO(4)/SU(2), then you do have to choose one of the “chiral” SU(2) subgroups. In this case, the quotient group is unambiguously SU(2)/{+/-1} ~ SO(3) with generators drawn from the “opposite” chiral subgroup. But that’s not the only way a quotient of Lie groups can enter into physics. If all you want is a finer fibration of a principal bundle with an SO(4) fiber, then it’s not important to have a group structure on the quotient space, and you can choose any Lie subgroup of SO(4) as the (sub-)fiber.

Peter’s later comment about an “anti-diagonal” SU(2) is indeed erroneous, in that the generators “orthogonal” to the diagonal SO(3) don’t form a closed Lie algebra. And for this reason, I think the “diagonal” SO(3) isn’t a terribly helpful starting point if you want to wind up with an SU(2) that you can identify with the electroweak SU(2)_L. But there’s another way to implement the spirit of his original post, which is to look at the manifold structure of SO(4) / (subgroup containing SU(2)_L) and ask what the action of the remaining generators on this manifold looks like.

This matters when you try to express, say, a Chern character on a 4-manifold M in terms of a more complex bundle over M than the ordinary frame bundle. Why would you want to do this? Because we have much more powerful analytical techniques for systems phrased in terms of complex fields than for those phrased in terms of real fields, and this more complicated bundle structure may contain a central U(1) that we can identify with the “i” of complex analysis.

For instance, you could take an individual chiral generator — call it I (in a convention in which the “generator” is the actual Lie algebra element and not the matrix I/i) — and consider the non-normal Lie subgroup G1 equal to the centralizer of I (isomorphic to SU(2)xU(1)/{+/1} ~ U(2)). We can’t put a Lie group structure on SO(4)/G1. But we can form a principal bundle with total manifold F ~ SO(4), fiber G1, and base manifold F/G1 ~ SU(2)/U(1) ~ S^2. The quotient is done along the orbits of the right action of G1, which also makes sense _globally_ on other objects with a global SO(4) right action — such as the SO(4) principal reduced frame bundle P of a 4-manifold M of arbitrary orientable topology. So you can look at the whole frame bundle as a principal G1-bundle E, globally diffeomorphic to P, over a bigger base manifold N=P/G1, locally diffeomorphic to (U \subset R^4) x S^2.

This 6-manifold N is in turn an associated SO(4)-bundle over M with fiber S^2, trivially reducible to an SO(3)-bundle since the generators of the SU(2) with opposite “chirality” to I (let’s call it SU(2)_L) all act trivially on the fiber. In general you can’t find a global section of this associated bundle, i. e., a smooth mapping from each point on M to a point on the fiber over M; if you could, then you could reduce the structure group of P to G1. But you can ask how a connexion on the original fiber bundle acts on the S^2 fiber of the associated SO(3)-bundle N and on the G1 fiber of the new principal bundle E.

We were able to reduce the structure group of N to SO(3) because both the left and right actions of SU(2)_L act trivially on its fiber S^2. So does the right action of the U(1) with generator I, as long as the quotient on SO(4)/G1 is taken from the right; but the left action of exp(It) on SO(4) doesn’t follow the same orbits as the right action. And because the fiber bundle structure of SO(4)/G1 is not trivial (it’s related to the Hopf map SU(2)/U(1) ~ S^2), we can’t reduce the structure group of E to SU(2)_L. Although the right action of G1 on an individual fiber of E looks like U(2), and we can certainly define a left action of the central U(1) on that same fiber for the purpose of constructing an atlas on E, this left action does not coincide with the left action of G1 as a subgroup of SO(4) on the original bundle structure P.

This complicates the mapping of fields of geometric origin on P to fields on E, which are (potentially) more analytically tractable. For instance, a connexion on the original fiber bundle P, viewed as a lift of the space of tangent vector fields on M to the space of right-invariant tangent vector fields on P, is more like a left action than a right action (in the sense that, if point p \in P has coordinates (x_i, g_i) in a given coordinate patch, and the right-invariant vector field v has the coordinate expression (v_i, g_i a) at p for some element a in the Lie algebra of G, then at R_g(p) = pg = (x_i, g) it has value not (v_i, g_i g a) but (v_i, g_i a g)).

So from the point of view of gauge couplings, i. e., the connexion on the original frame bundle, the original SO(4) splits into an SU(2)_L that acts on the fiber of the new G1-bundle and an SO(3) that acts on the “compactified” S^2; the latter also acts on the central U(1) of the G1 fiber in a way that varies depending on where you are on S^2. Except that’s probably not the right way to define a connexion on E that is in some sense induced from the connexion on P; instead, you want to go back to the idea of a connexion as a family of horizontal subspaces of the tangent spaces at various points of the bundle, and ask whether there is a consistent way to extend the 4-dimensional horizontal subspace at a point p \in P to a 6-dimensional horizontal subspace at the corresponding p \in E. The answer is yes; although the right actions of the other two SU(2)_R generators are associated with vector fields on each fiber of P which are not invariant under the right action of the U(1) in G1, the plane they span is.

Enough math. Looking back at Peter’s original idea, it is tempting to identify SU(2)_L with the weak SU(2) and seek a (not necessarily trivial) relationship between the central U(1) and the hypercharge U(1). As he proposed, there is also a resemblance that I won’t go into right now between the SO(3) action on S^2 and the SO(3) of spatial rotations on the original bundle; and analytical results (if any) will probably involve spinors and Wick rotations, not so much because the geometric objects sort neatly into spinor representations (they don’t) but because that’s a useful way to implement “gauge fixing” of the diffeomorphism group of M.

Does this have anything to do with the fundamental reality underlying the Standard Model? Maybe, maybe not. But it seems to me to be a perfectly sensible starting point for an interesting mathematical excursion. And having audited and enjoyed a string theory course not so long ago, that’s more or less how I feel about string theory, too. It’s just not science — not until it has some combination of predictive and explanatory power with regard to the observed universe.

Cheers,

– Michael

Michael, given Peter’s recent comments, your math filled post about Peter’s ideas might be more common in the future of this blog. That 4 in SO(4) is a Clifford Algebra vector and why more people don’t play with a vector for their spacetime is beyond me. If you want a ten dim spacetime then John Baez’s SO(10) paper seems like a nicer starting point than string theory. If you want to naturally get to spinors then SO(8) and its triality seem nice, even string theory noticed this years ago before getting farther and farther from the real world. One can also do nice things with SO(8) and fiber bundles. That some kind of vector-spinor triality might exist down at SO(4) is a nice idea. Finding little math errors does not always make the general idea wrong.

Kasper,

Go talk to any “string phenomenologist” working with the state of the art of techniques for trying to get physics out string theory. What they do is pick a background of some kind and try and calculate something. Different backgrounds give different results.

Dear Peter

Go talk to any âphenomenologistâ working with the state of the art of techniques for trying to get physics out of quantum field theory. What they do is pick a background of some kind and try and calculate something. Different backgrounds give different results.

cheers, Kasper

Michael,

Many thanks for the long comment. The kind of geometry you’re talking about is exactly what I had in mind in the comments you’re referring to. The “off-diagonal” SU(2) comment was a misguided attempt on my part to say something over-simplified about the geometrical set-up that you describe in detail.

The space you call N is also known as the (Euclidean) twistor space. a CP^1 bundle over the four manifold M. It’s the bundle over M whose fiber at a point is the space of orthogonal complex structures on the tangent space at the point (more precisely, those compatible with the orientation). As you note, the frame bundle P of M gives a principal U(2) bundle over N. It is this U(2) that I’d like to identify as the electroweak U(2).

Again, as you note, the problem with this is that this is a U(2) bundle over N, not M, and one has to deal with this somehow, and I don’t know of a really satisfactory way of doing so. You could just locally pick a section of N (in general you can’t do this globally), but then you have another sort of field to worry about. The SU(2) part of the U(2) doesn’t depend on this, but the U(1) does, which creates a problem.

One other issue here is that this is all Euclidean, and how to set up the analytic continuation to Minkowski signature has to be sorted out. In essence, the hope is that somehow under analytic continuation the boost part of the local Minkowski frame transformations becomes something that can be interpreted as an internal SU(2) symmetry.

My suspicion is that this somehow has to do with an old conceptual problem that many people have commented on (Yang, Penrose, Baez and others): there are several complex structures being used in QFT, and it is not clear why they can all be identified as acting by the same “i”. One kind of complex structure comes from space-time symmetry, it’s used crucially to distinguish between positive and negative frequency, and thus to characterize the vacuum state. Another inherently different complex structure is the one on the fibers, the “i” that generates U(1) gauge transformations. I’d like to think that maybe the use of the space of all local complex structures gives a new way of thinking about this problem, but I haven’t seen my way through this.

At some point in thinking about this many years ago I reached the conclusion that I needed a better, more abstract, way of thinking about path integrals, especially for fermi fields, before it would become clear if the ideas above could be used. Still working on that….

Kasper,

OK, so you agree with me that I was correct to write in my book that string theory depends on a choice of background.

You obsessively want to make the argument that string theory is in the same state as QFT, even though this is obviously not the case. One makes testable predictions, the other doesn’t. I’ve gone over this a hundred times here, and doing it the one-hundred and first time is clearly going to be a waste of time. But the bottom line is extremely simple to state: QFT phenomenologists can calculate many things that can be compared to experiment, and, if they don’t start making their models baroquely complicated, they make solid predictions that can be tested. String theory “phenomenologists” can calculate very little in their models, and even to force this very little to agree with experiment, they have to make their models so complicated they make don’t make testable predictions.

String theory does not depend on the choice of background. The low energy observed physics depends on the choice of a vacuum. Just like in any other theory with multiple vacua.

OK, Aaron, I knew you wouldn’t be able to help yourself.

You should mention that it’s not just the low energy physics that depends on the background, but the high energy physics also.

All known ways of writing down string theory depend on first choosing a background (yes, even AdS/CFT, where the asymptotic behavior is the background). At this point I recall, what you like to do is to say that string theory is just analogous to gauge theory, it’s just we only have a gauge-fixed version. Problem with this argument is that observable physics doesn’t depend on the gauge, whereas observable physics of string theory does depend on the background (which most people not schooled in string metaphysics would describe as being background dependent).

If you have something new to say about this, go ahead, but if it’s going to be the same argument, please just link to one of the versions elsewhere.

And who, don’t even think of joining in. Unless you have something really new to say on this topic, any comments arguing about this will be mercilessly deleted.

Peter,

I don’t know what you wrote in your book since I haven’t read it yet. So I can’t say that I agree with you.

And actually, it is not true that I “want to make the argument that string theory is in the same state as QFT”. Of course not. My question was not related to phenomenology directly. Actually I think it was quite simple:

Maybe you’ll just need to make more precise what you mean with the concept of “background independence”; And then the question was I don’t think you answered yet was:

So I don’t want to repeat the discussion of whether string theory makes any “real predictions” or not; the discussion would be much simpler if I had read your book đ

best regards, Kasper

Actually, what you say is not how I think about it, but as per your request, those interested can see here. Enjoy!

Dear Aaaron and Peter,

Of course the question about a choice of background is different from the one of a choice of vacuum.

My question (2) was related to Peter’s criticism of string theory in that you choose a background (much like in many other theories); but as we all know the resulting physics is background independent; so his critisism seems to be formulated in a way that confuses things.

My question (1) was related to the fact that the concepts of a choice of background and vacua also is confused; surely different points in the string theory landscape give rise to different physics – different cosmological constant etc. etc.; the background g’ = exp(2) g is equivalent to background g (as Peter agreed) and not a different vacuum, or related to the question of “giving different physics”.

Kasper

“as we all know the resulting physics is background independent”

You’re confusing what is known to be true with what many people would like to be true.

In any case, this is not even conjecturally true in the case of what people expect for generalizations of AdS/CFT. The string theory explicitly depends on a fixed asymptotic background. Different asymptotic backgrounds give different physics.

Dear Peter,

Sorry, maybe not all…. And, are my questions above answered in your book, NEW ??

Best regards, Kasper

Kasper,

No, the book doesn’t deal with every piece of wishful thinking common among string theorists. All it does is accurately describe what the current state of knowledge in the field is.

Yes, yes, I finally did get some more copies of the book recently, and you’re on a list of people to send a copy to…

Peter,

It’s great to have the “twistor” connection to the literature about this and similar constructions. I’ve read a bit about twistors in Minkowski space but hadn’t identified this particular construction with Euclidean twistors. Is there a good book in this area that focuses on geometrical and topological applications? (My idea of “good book” on manifolds runs to Goeckeler & Schuecker and Choquet-Bruhat; I learned differential geometry originally out of Dubrovin but was later turned on to differential forms and fiber bundles by Bill Burke and tend to prefer that language.)

What I like about the construction in terms of the frame bundle is that it makes clear that you can have a global U(1) right action on certain geometrical objects on any orientable 4-manifold, without postulating additional structure on the base manifold. It also shows how you can get “compact” dimensions with geometrical significance, in addition to the “macroscopic” dimensions of the base manifold, without a lot of handwaving about why some dimensions are macroscopic and others aren’t. Depending on what you’re trying to calculate, you may be able to work entirely on the U(2) bundle over N and quotient out the redundant degrees of freedom at the end, never having to deal explicitly with the lack of a complex structure on the original base manifold.

I do not worry about continuation to a manifold of intrinsic Minkowski signature because I am exploring the premise of complete diffeomorphism invariance, in which geometry is interesting only as a way to express topology in terms of local fields. The Minkowski signature isn’t intrinsic to the manifold; it comes of choosing boundary conditions on the set of acceptable coordinate systems in which t->-inf and t->+inf are fundamentally different from spacelike infinity.

Picture a 2-torus M covered, except for a 1-dimensional “skeleton” consisting of two intersecting circles, by a single contractible region. The maximal atlas on the 2-torus contains a diffeomorphism between some coordinate region U \subset R^2 and this region of M. Choose a diffeomorphism between U and the complex plane C, and remove the origin and the negative real axis. In terms of radial coordinates rho and phi, define t = log rho and x = tan phi/2. The entire t->-inf boundary of this R^2 coordinate system converges toward the origin; the x->+/-inf boundary converges toward the negative real axis; and the t->+inf boundary converges toward the “skeleton”. The t->+inf boundary is fundamentally different from the others; its intrinsic geometry contains “kinks” that capture the topological difference between the 2-torus and the 2-sphere. Euclidean rotations of this coordinate system do not preserve this difference but Lorentz boosts (pure shears along the light cone axes) do.

Now add two dimensions and a vastly more complicated topology. The premise remains the same: one R^n coordinate system covering all of an orientable manifold M except for a (n-1)-dimensional skeleton (actually, a simplicial complex of dimension not exceeding n-1) plus a trivial coordinate anomaly, in this case arising from recoordinatizing U->Q->R^4. (There’s nothing magical about the quaternions here; they’re just a convenient way to get to an R^4 system with the desired boundary conditions.) The skeleton is at t->+inf, and t->-inf converges to a single point on the manifold, connected by a line segment to a point on the skeleton; all of spacelike infinity winds up on this line segment.

None of these boundaries are special in terms of the intrinsic geometry of the manifold; one could just as easily have picked any other maximal coordinate patch in the atlas on M and transformed it using the same U->Q->R^4 trick. But the t->-inf and t->+inf boundaries are certainly special in the coordinate system, and the Poincare group preserves this distinction. I consider this quite sufficient reason for a completely diffeomorphism invariant theory to have the global causality structure and Poincare-invariant phenomenology with which we are familiar from QFT. And for dessert we get primordial asymptotic homogeneity and an entropic arrow of time.

Remember, I’m not saying this has anything to do with physics. This obviously isn’t the whole spacetime story; it doesn’t address microcausality, it doesn’t explain why gravity looks so much like intrinsic geometry on a Minkowski background, it doesn’t tell you the price of tea in China. But it adds to my interest in in Wick rotation as a way of making contact between QFT actions and partition functions in Euclidean space. Not to mention an interest in finding the portion of the SM gauge spectrum that couples chirally somewhere in the frame bundle over a Euclidean 4-manifold.

I also have some thoughts on path integrals and fermions, which it will take me a bit longer to write down at finite length. They were inspired by the identification of the Faddeev-Popov ghost field with the Maurer-Cartan form on the group of QCD gauge transformations. I am handicapped in expressing these ideas by an inadequate grasp of the theory and lingo of Virasoro representations, which seem to be the way that field theorists with a solid mathematical background look at this. Any recommendations on texts?

Cheers,

– Michael

Of course, I meant H for the quaternions. I had originally written C^2, changed it during an editing pass, and had a momentary brain lapse. I’m a bit out of the habit and not copy editing as carefully as I might (there are glaring typos in the earlier post too).

– Michael

Michael,

I learned about the Euclidean version of the twistor construction from Atiyah. He used it right at the beginning of the modern interaction between math and physics to study instantons. Using it relates questions about self-dual connections to questions about holomorphic bundles. See his short book “Geometry of Yang-Mills Fields”, in vol. 5 of his collected works.

The Virasoro algebra is the centrally extended lie algebra of diffeomorphisms of the circle. There’s a huge literature about this by now. The group of gauge transformations is something different, but related. For the circle, it’s a loop group, with the central extension the Lie algebra is an affine Kac-Moody Lie algebra, and there’s again a huge literature. For spaces of higher dimension than a circle, very little is known about the representations of either gauge groups or diffeomorphism groups. One place to start reading about this is Jouko Mickelsson’s book.