Open System for Geniuses - by Chronostalker

Let’s continue from Part 1. We are discussing, critically, a paper "Quantum Field Theory and Representation Theory: A Sketch" by Peter Woit. The paper is also available from his website. This must be not a very popular paper as the Citebase reveals only 0 citations and 16 downloads of the paper. For a comparison, the PhD Thesis of Lubos Motl, "Nonperturbative Formulations of Superstring Theory", available from http://arxiv.org/abs/hep-th/0109149, shows 0 citations and 53 downloads, while "Higher Yang-Mills Theory" by John Baez shows 19 citations and 29 downloads (these downloads numbers are, as it seems, only from uk.arXiv.org ).

We continue commenting, sentence by sentence, the introductory part of the paper. Peter Woit writes:

The relation between quantum mechanics and representation theory has been formalized as the subject of \lq\lq geometric quantization" which ideally associates to a classical mechanical phase space (a symplectic manifold M) a complex vector space V in a functorial manner. This functor takes at least some subgroup $G$ of the symplectomorphisms (canonical transformations) of M to unitary transformations of V, making V a unitary $G$-representation.

While the above is true, it is not the whole truth, nor even an essential part of the truth. Every symplectic manifold M is naturally endowed with the canonical volume form, and therefore also with a natural measure, let’s call it m. Symplectomorphism are automatically measure preserving transformations. We can always build the Hilbert space of square integrable functions on M. Then all the symplectomorphisms are represented by unitary transformations, and the construction is certainly functorial. In geometric quantization we aim in "cutting this Hilbert space in half", otherwise the task described in the quoted paragraph above would be trivial!

The theory of geometric quantization has never been very popular among physicists for at least
two reasons (in addition to the fact that the mathematical apparatus required is rather
extensive and mostly unfamiliar to physicists). The first is that it seems to have very little
to say about quantum field theory. The quantum field theory of the standard model of particle
physics is built upon the geometrical concepts of gauge fields and the Dirac operator on
spinors, concepts which have no obvious relation to those used in geometric quantization.

Again, that is not exactly correct. The problem of the Dirac operator and spinors arises in relativistic quantum mechanics, without the need of going to quantum field theory. There exist generalizations of geometric quantization that unable us to deal with spinning (free) particles. Studying the Poincare groups, its Lie algebra, and using an appropriately adjusted coadjoint orbit methods one gets a sort of relativistic quantum mechanics. The problems arise when we want to have a consistent physical interpretation, but these problems arise already in a non-relativistic case.

The second problem is that the most well-developed formalism for doing calculations within
the standard model is the path integral formalism. While no rigorous version of this exists,
all evidence is that consistent calculations can be performed using this formalism, at least
within perturbation theory or outside of perturbation theory with a lattice cut-off. Even in the
simple case of quantum mechanics the relationship between the path integral quantization and
geometric quantization has been quite unclear making it impossible to see how the ideas of
geometric quantization can be useful in the much more complex situation of standard model
quantum field theory.

I do not think that the fact that there is no evident relation between path integral formalism and geometric quantization constitutes a problem. There is no evident relation between path integral formalism and calculating of eigenvalues and eigenstates of operators, and yet physicists calculate these eigenvalues and eigenvectors for ages for a simple reason that these calculations seem to be useful. If physicists would find geometric quantization "useful" - they would use it disregarding the lack of its relationship to the path integral methods.

To be continued

Chronostalker

Today I started reading, out of simple curiosity, a rather long paper by paper Peter Woit, "Quantum Field Theory and Representation Theory: A Sketch." The paper is available from arxiv.org at the URL: http://arxiv.org/abs/hep-th/0206135 . 56 pages is a lot, especially when packed with data. But the subject is interesting and also somehow related to the other material discussed in my blog. I thought I will be picky, like others are picky when jumping on Bogdanovs. But right after starting reading Peter’s masterpiece I realized that it will not be that easy to be picky! Indeed, right at the end of the Introduction, just before starting the main part of the text, the author writes:

"The reader is to be warned that the present version of this document suffers from sloppiness on several levels. If factors of $2$, $\pi$ and $i$ seem to be wrong, they probably are. Many of the mathematical statements are made with a blatant disregard for mathematical precision. Sometimes this is done out of ignorance, sometimes out of a desire to simply get to the heart of the matter at hand. The goal has been to strike a balance such that physicists may have a fighting chance of reading this while mathematicians may not find the level of imprecision and simplification too hard to tolerate."

If only Bogdanovs were that smart! They would write in their papers and in the introduction to their book: "The reader is to be warned that the present version of this document suffers from sloppiness on several levels." But they were not, and so their critics rightly complain about their "blatant disregard for mathematical precision." Jokes aside, at least it seems to me that I can understand Peter’s ideas, and so to disgree with him, or to criticise his statements, gives some kind of a satisfaction. So I will be only moderately picky, because how can you be mad at someone who is humble ("sloppiness on several levels") the way Peter Woit is…..

I do not know yet neither how far will I go, nor which part of the paper I will focus on. Time (which, as we know from Connes and Rovelli, is the one parameter group of automorphisms of some algebra, the group associated to a stationary equilibrium state of a something) will tell. I will move slowly, step by step, pausing here and there if I see something that attracts particularly my attention.

Let us start with the introductory section.

Ever since the early days of theory there has been a close link between representation theory and quantum mechanics.

This seems to be correct. However, this is only true if we will consider Mackey’s theory of "imprimitivity systems" as a part of the representation theory. In quantum mechanics we are dealing with imprimitivity systems all the time, while groups enter only occassionaly, in special cases. It is true that Mackey’s theory of imprimitivity systems is important for the theory of group representations, but it can not be replaced by the theory of group representations - unless we move into the realms of infinite dimensional groups of maps - where the ground becomes shaky. I will return to this point pretty soon below.

The Hilbert space of quantum mechanics is a (projective) unitary representation of the symmetries of the classical mechanical system being quantized.

OK. Here we have the first sloppiness. What a pleasure for a referee of a paper to find such a pearl! The Hilbert space is NOT a representation! It carries a represenation! (Refereess love finding such totally inessential "errors". Correcting them proves that they really read the paper!) But even when read correctly the statement is not quite correct. The key problem is: how do we define a "symmetry of a classical mechanical system"? For instance, when we use a Lagrangian formalism, do we assume that the Lagrangian must be invariant? Or can it be invariant up to a divergence? Etc. Etc. And what if the classical systems is such that has no symmetries at all? What sunitary representation are we talking about in that case?

The fundamental observables of quantum mechanics correspond to the infinitesimal generators of these symmetries (energy corresponds to time translations, momentum to space translations, angular momentum to rotations, charge to phase changes).

Again incorrect statement. The fundamental observable in quantum mechanics is the position. Some authors go even that far they claim that it is the only observable, and that all other observables can or should be expressed in terms of positions. Bohmian mechanics, for instance, is all based on such an assumption. Path integral formalism works well in the configuration space, but needs special care when we move to the "momentum space", not to mention the fact that the very concept of the "momentum space" in general relativistic quantum mechanics asks for a serious attention. It is true that in case of a Galilei invariant quantum system the position observable happens to be mysterously related to a boost generator. But when the system is not Galilei invariant, when there are external forces, or when we move to special or general relativity, the position becomes the fundamental observable, independent of any group considerations.

To be continued

Chronostalker

In a comment to his blog Luboš Motl reveals his agenda. He writes there:

[…] I believe that this is one of the main reasons why the LQG people are the loudest critics of Bogdanoffs. The LQG people simply do not like the idea of a mirror that shows their work to be more or less equivalent to something that two French science-fiction writers can compose within a year. But it more or less is, and as far as I am concerned, the speculation that the initial singularity should be described by a phase rooted in topological QFT is a much deeper idea than the idea that the spacetime should be made of links and vertices.

Whether the initial singularity (if there were such singularities at all) is better described by topological QFT or by the idea that the spacetime should be made of links and vertices - is yet to be seen. Perhaps both ideas are utterly wrong. Future development of physics will show (or not). Notice that Luboš has started his blog on “Bogdanoff paper” by stating that Bogdanovs are ” two French journalists and scientific comedians”. This is evidently derogatory. Technically Bogdanovs are as much “scientists” as Luboš Motl - they have legitimate PhD’s and publications in scientific journals. Luboš Motl, on the other hand, can be described as “scientific comedian” - see his web page “Home page of Abbe Hyupsing Qong”at http://www.physics.rutgers.edu/~motl/. Yet he is not talking about himself as a “comedian”. This is unfair. Similarly unfair is the following comment by Lubos:

[…] I apologize but I had to erase two comments - a speculative thread started by an anonymous author - because it was too sensitive. Please don’t be afraid do reveal your identity. It may be unpleasant if someone (scientists in this case) is kind of attacked by a writer whose identity is unknown.

Yet when Bogdanovs, who are also scientists, just somewhat different, were wildly attacked by writers whose identity is unknown - Luboš did not protest!

Back to the comparison between Bogdanovs papers and LQG. There is, I think, an essential difference here. LQG papers, like many papers in theoretical physics, may have pieces of some healthy mathematics - except that it is useless for physics. And the authors, very often, have very shallow, wrong, or not at all understanding of physics. Yet they use physics terms - because in this way their papers get published and they think that they are doing “physics”. Bogdanovs papers, on the other hand, seem to have interesting ideas and healthy sense of what may be relevant for physics, yet they have severe holes in their mathematical education. I have good reasons to beleieve that they do not know what is a tensor product of two vector spaces, what is a contravariant tensor and how it differs from a covariant one, what is a fibre bundle (apart of the fact that it has a base space and fibres ), etc. They certainly do not know the definition of a von Neumann algebra and they probably think that they do not need to know - you can speak of KMS states without really knowing what they are. So we have two different categories of a “boundary science”. We have papers like the one by Connes and Rovelli, presenting correct though trivial mathematics and showing complete shallowness concerning physics, and we have Bogdanos who are shallow or even wrong in their mathematics, but who may have a good sense as to which areas of physics may need more attention. One should not discard these boundary papers. The paper by Connes and Rovelli, even though shallow and sometimes even wrong, may be very useful for someone to produce something better. And the same for the papers by Igor and Grichka Bogdanovs (or Bogdanoffs - if you wish) . The same with Cramer’s TI of quantum mechanics. Often wrong - yet by simply understanding what is wrong there , one can learn a lot! What is important is an open discussion and open, healthy, criticism. But criticism of details. Not putting labels on people or their work.

Answering “Quantoken” Peter Woit wrote:

[…] The Bogdanov papers are different. There are huge leaps of logic from one sentence to the next, clear examples of misunderstandings of basic ideas by the authors, etc. The problem isn’t their assumptions, but that they can’t construct a coherent argument and don’t understand the technical tools they are using.[…]

The last sentence fits the conclusions stemming from the dialog in The Bogdanov Affair - The Dialogue Continues… It is clear from this dialog that the brothers do not know the precise definitions of the terms that they are using, neither they have a real understanding of what a “mathematical proof” should look like. Yet Peter Woit is not quite right when he writes that “The Bogdanov papers are different. ” There are many, even eminent, theoretical physicists who are using the terms that they would be unable to define, like “manifold”, “topology”, even a “group” or a “spinor”. While it is difficult to detect such cases by just reading the papers (an so, it is difficult to blame the referees for letting such papers to be published), it is relatively easy to detect such a phenomenon in a dialog. An examiner, or a PhD referee has a duty to start a dialog when things that are “suspicious” show up. By asking simple questions and by getting avoiding answers, one can rather soon to come to the conclusion that the other person is ignorant in this or another area. The next question that can be asked is how far the ignorance goes. A couple of additional questions can reveal that level too. Yet….
Being “ignorant” is not necessarily “bad”. Sometimes good, innovative, ideas come ONLY because the author is “ignorant” in some areas. What to do then? I suggest to open a special section of arXiv.org and to start a new journal for “Impressionistic Physics”.

On June 16, 2005, Luboš Motl discovered the wheel and has been termed a crackpot by Peter Woit. In his blog, Luboš Motl’s reference frame, Luboš Motl added an entry ” The Bogdanoff papers”. He makes several not quite correct statements there. He writes:

“.. two French scientific comedians with Russian names, namely Gritchka Bogdanoff and Igor Bogdanoff (whom the French TV audience knows as geniuses from a certain TV show), published something in “Classical and Quantum Gravity” that the journalists promoted as the ‘reverse Sokal hoax’.”

That is incorrect. It was John Baez, not some “journalists”, that promoted the term “reverse Sokal Hoax” Journalists joined later on. Then Luboš writes:

“A very detailed summary of the affair was written by

* John Baez

but others have written comments about it, too - for example Peter Woit or Jacques Distler. “

Here Luboš forgets that the most detailed commentary has appeared from the very beginning of the whole affair on the site of Arkadiusz jadczyk. Luboš thinks that his blog article is one of the first ones that suggests that Bogdanov’s paper may look as good as papers of many experts. In this respect he is not well informed. He should read the comments by Daniel Strenheimer and Robert Coquereaux on Jadczyk’s site.
Peter Woit writes in his blog:

[…] Maybe the only scandal here was the laziness of referees, not the infection of the whole subject by nonsense to the point where lots of people can’t tell the difference.[…]

It is not just the laziness. It is their lack of responsibility, on top of laziness. It is also a serious error of the thesis’ director. And, it is also a general degradation of values within the community of theoretical physicists. “Impressionism” became a new style, and “impressionists” deserve specialized journals, where “the correct jargon” and “new ways of putting the stuff together” will be enough to get published. No rigour or understanding of the terms used will be needed there. Then the editors and referees of the old-fashioned, “realistic”, journals will simply reject “impressionistic” papers and suggest sending it out elsewhere, without judging their content. I will be sending most of my papers to the old fashioned journals, but once in a while I may write something in an impressionistic (or even surrealistic) style as well.