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.”

OK, then perhaps Virasoro representation is not the hook into the literature that I’m looking for. What I’m really after is a way of describing the Lie algebra cohomology of the diffeomorphism group of a compact orientable 4-manifold.

There’s an elegant way of combining the BRST coboundary operator with the covariant exterior derivative relative to a fixed connection to get a covariant coboundary operator on the gauge bundle. This can be used to construct a Lie algebra of graded derivations in which the “scalar functionals” are the space of Lagrangians (viewed as polynomials in the field degrees of freedom and their derivatives at a point, as in Ward identities) and the inner derivatives are taken relative to the direct sum of the Lie algebra of tangent vector fields on the base space and the Lie algebra of infinitesimal gauge transformations. The Maurer-Cartan form on this space is quite interesting.

I shouldn’t attempt to spell this out in more detail without LaTeX, but does it sound like one of the directions that cohomology has taken since Stora? Does it have anything to do with equivariant cohomology?

Cheers,

– Michael

What I’m really after is a way of describing the Lie algebra cohomology of the diffeomorphism group of a compact orientable 4-manifold.To simplify things, one should start infinitesimally and locally. Extensions of the algebra of polynomial vector fields by modules of tensor fields were classified by Dzhumadildaev – my review math-ph/0002016 is online. Two of the extensions are closely related to the higher-dimensional generalizations of the Virasoro algebra, which arise in lowest-energy representations.

Michael,

There’s work by mathematicians on the Lie algebra cohomology of vector fields that goes under the name “Gelfand-Fuks cohomology”. Bott wrote some beautiful expository papers on the subject, see vol. 3 (I think, the one on foliations) of his collected works.

The BRST related ideas you mention are among the things about this I’ve always found confusing and have never quite sorted out for myself, especially the relation to equivariant cohomology.

Definition of Gelfand-Fuks cohomology can be found here.

A clarification: an extension of a Lie algebra L by its module M is an element in H^2(L,M). Dzhumadildaev (Z Phys C 72 (1996) 509-517) classified this for L = vect(n) and M a tensor module. Gelfand and Fuks only considered n = 1 and M the trivial module (Gelfand-Fuks cocycle = Virasoro algebra).

TL, there should be a way of saying this physically, as in, “the part of curvature not coupled to matter is the Weyl conformal curvature” etc. There may be an obvious intepretation I’m ignorant of..

-drl

DRL. Sorry, but I specifically tried to answer a question on Lie algebra cohomology, without claiming any connection to physics.

OTOH, demanding a kinematical Hilbert space with a well-defined action of the diffeomorphism algebra dictates that QFT must be modified in a certain way; details can be found in the ArXiv. Unfortutely, what is written reflects my understanding as of 2004 and is flawed in several respects. If I stop wasting my time reading blogs, I might eventually manage to write things up.

Thomas,

Your ArXiv papers are definitely proving rewarding reading. There is, however, an aspect of your handling of the BRST operator that is not clear to me.

A textbook description (e. g., section 12.3 of Goeckeler & Schuecker) of Stora’s solutions to the Wess-Zumino consistency condition exhibits a linear representation W on the space Pl of Lagrangian polynomials of the total Lie algebra E of right-invariant vector fields on the gauge bundle. On a trivial gauge bundle, E is the semidirect product of the vector algebra and the gauge algebra over M. On a non-trivial gauge bundle, infinitesimal gauge transformations still form a “vertical” ideal of E, but the “horizontal” vector algebra does not form a subalgebra. The coordinate expression of the Lie bracket on E necessarily involves a “fixed” (background) connexion on the gauge bundle, but that’s not fundamental to the construction; it’s just a way of expressing a right-invariant vector field on the bundle space in terms of a vector field on the base manifold plus an infinitesimal gauge transformation.

Identifying Pl with the space \Lambda^0 of 0-forms on E taking values in Pl, the Ward operator W(e) may be identified (up to a sign) with the Lie derivative wrt e on \Lambda^0. This Lie derivative looks exotic, involving a lift of the vector algebra (the infinitesimal diffeomorphisms) from the base manifold to the gauge bundle by means of a fixed connexion, ensuring that the commutators in the Lie bracket on E can be patched together on the overlaps of a local trivialization. This Lie bracket should not be confused with the Lie bracket of vector fields on the base manifold; the curvature of the fixed connexion appears in the Lie bracket of two horizontal algebra elements.

Now, given a Lie bracket on the total algebra E and a Lie derivative on the space of Pl-valued 0-forms over E, we can axiomatically construct the complete algebra \Lambda of alternating forms on E and the graded Lie algebra of derivations relating them. The “exterior derivative” operator d_E in this construction can be identified (up to a sign) with the BRST coboundary operator, and in fact the 1-form d_E L coincides with -Q L on the vertical ideal of E. d_E is nilpotent by construction.

Now, is -d_E the Koszul-Tate operator referenced in 0501043? (I’m having a hard time relating the language of canonical quantisation to the functional setting in which I’m used to seeing BRST-related constructions such as the Faddeev-Popov ghost.) If so, how does your approach to quantisation result in a change to the BRST operator that breaks nilpotence?

Cheers,

– Michael

More notes on the above, in hope that one or more of the folks who have been so kind as to suggest readings in cohomology will recognize this as a standard line of reasoning.

It turns out that most of the above construction is available in the first five sections of Schuecker’s 1987 paper:

http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.cmp/1104116716

His construction of the graded Lie algebra of derivations in section 6 is, however, significantly different from what I have in mind. He also doesn’t particularly emphasize that the fixed auxiliary connexion is only a device for identifying the “horizontal” portion of an algebra element in a particular local trivialization, and that the actual graded Lie algebra of derivations on \Lambda is independent of this connexion. Nor does he spell out that the full coboundary operator (in his notation, d + s) is an extension of the BRST coboundary to the full algebra of infinitesimal diffeomorphisms + infinitesimal gauge transformations; maybe in his construction it’s not (I haven’t checked).

Perhaps for these reasons, Schuecker points out the resemblance of the “(algebraic) Faddeev-Popov ghost” to the Maurer-Cartan form but does not extend it to the full Maurer-Cartan form on E. The latter has some very interesting properties that I may comment on later (when I decipher some of my old notes).

In my construction, the Lie derivative on \Lambda with respect to an element e of E is of degree 0, the inner derivative with respect to e is of degree -1, and the exterior derivative (coboundary) is of degree 1, just like in ordinary differential forms. The inner derivatives with respect to elements of E form a Grassmann subalgebra of the graded algebra of derivations on \Lambda, which I am tempted to relate to the Grassmann algebra of which fermions carry odd representations.

I hedge a bit here because, if this has any relationship to physics, the fermions of low-energy phenomenology probably are not objects of a single rank in this Grassmann subalgebra. There could be some terms in the “fundamental” Lagrangian that arise from topological densities, expressed using objects of well-defined rank. Other terms are artifacts of functional quantisation in a non-diffeomorphism-invariant “gauge”, involving fields whose algebraic properties (like those of the traditional Faddeev-Popov ghost) are chosen so that the added term in the Lagrangian forms an operator trace of the Jacobian of the gauge-fixing term. I would expect the eigenfields of the mass term in the effective Lagrangian (wherever it comes from) to be a mix of fields of these two types.

Is this clear enough to be boring yet?

Cheers,

– Michael

Michael, it might be better to continue this off-topic discussion privately, rather than pushing Peter’s hospitality further. My email address is on my eprints. However, let me just end with some general comments.

What I’m doing is not strictly equivalent to QFT. To obtain a well-defined action of the diff algebra, which is impossible in QFT proper, I first replace all fields by their Taylor series. This introduces an additional datum: the expansion point. Whereas infinite Taylor series are independent of the base point, truncated ones are not, and this dependence remains in the form of anomalies after quantization, even when the truncation is removed.

The existence of new anomalies shows that passing to Taylor data makes a substantial difference. I think this is a good thing, because we know that QFT is incompatible with gravity. By considering a structure which is close to QFT, but essentially different from it, this no-go theorem might be avoided.

Our main similarity is that we both use cohomology, but this is a very general mathematical technique, applicable in many situations. My work is largely modelled on the antifield approach, as formulated in chapter 17 of Henneaux and Teitelboim. However, since I want to do canonical quantization, I need an honest Poisson bracket rather than an antibracket, and therefore my starting point is not the space of histories, but rather its phase space. A flaw in my paper is that I get an overcounting for the harmonic oscillator. To correct this, one must add an extra constraint which identifies momenta and velocities.

Thank you for making me aware of Schuecker’s paper. I have never managed to understand them properly, and I have long been confused about their relation to my extensions. AFAIU, there is none. In particular, I do consider vector fields on the base manifold, without a reference connection A^0.

to understand them properlythem = conventional gauge anomalies

Thomas,

My (obviously very amateur) take on anomalies is that they happen when you try to write down a theory about objects that live in quotient spaces of group actions on fiber bundles using index-slinging notation and you don’t get it quite right. This tends to come of using heuristics like “give this symbol algebraic properties which make the term in which it appears come out gauge invariant”, “the horizontal element is the one normal to the vertical subspace”, or “this field must be a boson because it’s a Lorentz scalar” instead of sweating blood over what it’s doing there in the first place.

We’ve all grown up with this situation because quantum mechanics _works_ even though it doesn’t make any bloody sense. Heck, it started long before quantum mechanics: I lost confidence in the formal correctness of what I was doing in the physics classroom the first time I saw a Lagrange multiplier, and to this day I can’t look at a partial derivative without wincing. The Lie derivative _means_ something. A partial derivative has no more intrinsic meaning than ten in the ones column carried to one in the tens column.

Until you slog your way through to BRST quantization, gauge covariance is just a heuristic. For me at least, putting Schücker’s chapter on anomalies side by side with Peskin & Schroeder’s explanation of “BRST symmetry” led (eventually) to an “a-ha” moment: if you want a theory with local causality, and the fundamental objects of your theory live on a big honkin’ principal bundle with irreducible global structure, then you’d better define your theory there, because that’s the _only_ space on which the Lie derivative looks like a local operator. The rest of the apparatus – differential forms, Clifford algebras, Wick rotation, Feynman diagrams, contour integrals – is long division for postdocs.

Your work has my respect (though not yet my comprehension) because it investigates how real calculations with imperfectly known initial conditions can be truncated without losing the answer in imperfectly cancelled anomalies. I couldn’t compute a QCD background if my life depended on it. A formalism with all the elegance and explanatory power in the world is of little use if it can’t make contact with phenomenology, and “we can’t prove that the SM can’t be obtained as our effective field theory” is a poor second to a theory with added predictive power. Personally, I would be very satisfied with a theory that doesn’t make any “new” predictions but does liberate grad students from index-slinging and the rest of us from piffle about the philosophical implications of discontinuous spacetime at the Planck scale.

Cheers,

– Michael