The Bogdanov Singularity

New - June 5

Towards the theory of matter, geometry and information (from the upcoming book)

Summary: This page contains a discussion of technical details of the papers by Igor and Grichka Bogdanov. It contains technical terms and it involves certain jargon that physicists (in this case: mathematical physicists) use in their discussions - usually with chalk and blackboards.

Neither the subject, nor the issues involved, nor the conclusion will be clear to a non-specialist.

Therefore I decided to write this introduction that will tell the actual status of the story in plain terms to the general reader. When dealing with a subject like one we are dealing with saying "it is all nonsense", or "it all makes no sense to me" is misleading, cheap and highly inapropriate. The devil is in the details.

If something does not make sense to me - even if I consider myself an expert - it does not mean that the same thing will not make sense to another expert.

So, how to proceed?

In my own experience as a physicist I once experienced a similar situation. A younger collegue was having problems with his papers being accepted for his PhD. Somehow his papers and seminars were so difficult to understand that the faculty could not decide whether what he was doing could be considered sufficient to start the formal procedures, or whether it was all a "hoax."

Because of my wide interests and expertise, the faculty decided to rely on my expertise. Because I like to be challenged, I accepted the job of examining the details in order to arrive at a final conclusion about the matter.

After the first reading and first few hours of discussion with the young man, my working hypothesis was: it's all nonsense. The guy does not even understand the terms he is using! But I like to be sure.

It so happened that I also had gaps in my education (no physicist or mathematician does not have such gaps), and not all the terms involved were 100% clear to me. The subject is not important, but for those who might like to know: it was about representations of CAR generated by representations of CCR, in particular non-Fock representations. The Mathematics involved a theory of continuous and incomplete tensor products developed by John von Neumann. So I decided that in order to find the errors I really had to become an expert in all areas involved.

I questioned the candidate for hours, trying to get from him the real meaning that he was trying to formulate. I then spent hours alone trying to figure out if there was a way in which his idea could be precisely expressed, as well as finding the conditions under which the "results" could be mathematically proven.

I worked ten hour a day for more than a month on this problem. What was the result?

My thinking evolved from: "it's all nonsense", and "it doesn't make any sense, plainly wrong" to "perhaps something valuable can be extracted" ; and then to "perhaps one can even get some new results...".

The final result was: our candidate's completely rewritten papers were published in respected journals, and he was awarded his PhD. He tried to convince me that I should receive credit as a co-author of one of the essential papers, but I flatly refused - as far as I was concerned, I had just "helped".

From that experience, I learned a valuable lesson: that there is a style in theoretical physics that may be called "impressionistic". It is true that I would really appreciate if each "impressionist" would start his paper with:

"I am an impressionist. Although I do not fully understand the terms I am using in this paper, still I worked really hard and I would appreciate any constructive or even destructive criticism of my work. My list of refences includes, I believe, all important publication on the subject, and I really studied them and did my best trying to understand what they are about. If I missed something or missrepresented something - please, let me know, and I will correct asap. And here are my own ideas: ... "

Something like this would make the life of the more practical physicist so much easier! But it's hardly likely that journal redactors would accept such a sincere statement. Journal editors easily accept papers that nobody (especially referees) can understand. But a sincere paper? Never!

Now, back to the case at hand: "The Bogdanovs Affaire ( or Bogdanov Affair )". After some digging, corresponding with Bogdanovs and with fellow physicists, after re-reading several times their publications, it seems to me that their case may be similar to the case I have described above. That is, after constructive input and help, something rather interesting can be extracted from their work, and that this something will bear the stamp of Bogdanov's out-of- the- box, unrestricted by the standard of a priori rigour, thinking. At present, as of November 6. 2002, it is just my new working hypothesis. Whether it will be confirmed or refuted - depends to a great extent on the constructive input from the experts in the field. In particular on my own input.

The Reader may wish to know what are my own qualifications. It so happens that my own PhD was on the subject of KMS states, that my two papers on von Neumann algebras with cyclic and separating vector, one dealing with statistical mechanics and one dealing with algebraic quantum field theory, were published in Communication in Mathematical Physics, that they are quoted in the monograph on Algebraic Methods in Statistical Mechanics and Quantum Field Theory by Gerard G. Emch (UFL), and also in the monograph on C* algebras by J. Dixmier (this last fact is not a surprise as the monograph has >2000 references!). I also collaborated (and co-authored papers) with Daniel Kastler - one of the few pioneers in the field. Last but not least I was a Director of the international school of theoretical physics on the subject "Fields and Geometry" - so I know quite a bit about many parts of Bogdanov's work.

From a letter from one of the permanent members of Laboratoire Gevrey de Mathématique Physique, Université de Bourgogne:

Dear Arkadiusz,

i have been following this story on the news groups after some string physicists alerted me that there were rumors that the Bodganoff brothers were professors in Dijon. Actually they are not affiliated with our laboratory (they were when they were preparing their thesis).

I attended the PhD defence of Igor, and although I am not an expert in that field, it was obvious to me that his (their) work was kind of impressionistic and didn't make sense to me. Somehow, they managed to get these "impressionistic" papers published in CQG and Annals of Physics by impressing the referees (both journals and thesis), and it is still a mystery for me how they could appear in these journals. For sure, this affair has created a crisis in the string/quantum gravity community, but there is no hoax here! Apparently, the brothers were really serious about their work.

Daniel (as administrative advisor) ... relied on the positive reports received from well-established experts. I think that's natural since both theses were out of his field of expertise. [...] (...for the administrative side, how one can refuse a student to pass his thesis, after he has been published and received positive reports from the "rapporteurs"?).

The positive point in this story is the sociological aspects that it uncovered: first it was a hoax, then people thought that they were academics and edulcorated their judgements, and finally started a kind of "community auto-critics" about the level of papers published in that field.

I guess that there is more to come...

best regards ...

This morning (October 30, 2002) Ark received a call from Daniel Sterheimer, "Directeur de Recherches au CNRS" at Laboratoire Gevrey de Mathématique Physique, Université de Bourgogne, Dijon. Daniel shared with us the background of the "Bogdanovs' affair" which has, apparently, been going on since 1992.

The story - in brief - is that there have been numerous changes of the theses subjects, writing and rewriting again and again; attempts to get published in various journals; changes of supervisors and referees; internecine struggles between different institutions of science; political pressures applied from various directions, and other dramatic events that - if fully chronicled - would make a great movie! (Daniel has promised to write a short summary of the essential points that we will present as soon as it arrives.)

The sociological milieu of the affair is rich in color and personalities. The Bogdanovs have been described as "charismatic" and and "persuasive." One of the juicier details is that the Bogdanov twins actually have FANS - scores of beautiful, nubile young women - who attend their seminars thereby delighting the other physicists - distracting them from theoretical to applied pysics. The Bogdanovs have political cachet - even the support of the French Minister of education. In addition to this, because of their prominence, the Bogdanov brothers have brought various publishing houses into the fray, making for a general circus-like atmosphere in the normally subdued and monkly Ivory towers of science.

Nevertheless, as far as we can determine in our investigation, what emerges is that the Bogdanovs have worked really hard within the window of opportunity created by the "impressionistic style" of contemporary publications in the field of string theory etc. They obtained their PhDs with a score that alllowed them just that - to pass. The fact that their papers have been published is also due to the fact that similar papers ARE being published in the field. If the Bogdanov's have used some concepts that they themselves do not really understand - that is nothing new in the field. In fact, it is pretty standard.

Over the past 50 years there has been a steady and exponential degradation of the process of education in the field of physics and mathematics that is not merely deplorable, but could actually be seen as a dangerous thing considering the world we live in and the forces that physics has unleashed thus far. Daniel is fond of telling about an experiment he conducted which demonstrated that an attorney, who graduted 40 years ago, could answer a simple but tricky algebra question better than a brilliant student of the exact sciences today. What's more, the student in question was son of an eminent physicist! Daniel (and others) hope that the Bogdanovs, with their fresh, innovative approach, as well as their broad spectrum relations in the real world, will pump new blood into a dying profession, popularizing physics, and most especially demonstrating that the adventures of the mind are the greatest challenges of all.

According to Daniel, Moshe accepted the Bogdanov's under his supervision mainly because there were so many others who were rejecting them because of their dynamic approach. Moshe felt that freedom to think out of the box was crucial to science - and we most definitely agree - but unfortunately Moshe did not live long enough to take care of the details. Daniel, taking over after the sudden and tragic death of Moshe, did the best he could, relying, as we have said, on the "expertise" of the referees. There is no conceivable fault on Daniel's part in concluding the business as he did.

The thesis of Igor Bogdanov seems to be - upon initial inspection - an ingenious mixture of mathematical jargon and physical jargon. According to Daniel, when Igor first presented his thesis, it was over 600 pages in 9 point type. So, nobody can say that a lot of work didn't go into it! However, the work of refereeing a 600 page paper was a challenge which few physicists are willing to accept. Apparently there was no single person who could be used as a referee for the whole thesis. Roman Jackiw of MIT is mentioned as the referee for "the physics part" of the paper. This suggests to us that Roman Jackiw did not check for mathematical inconsistencies. It is plausible that nobody did. In any event, Jackiw - as the referee - bears the burden of responsibility for the thesis being accepted. His name - and the institution behind him - is one of the "biggest" in the list of names of referees. Without Jackiw, it seems, there would be no thesis, no Ph.D., no claims of "hoax," no discussion at all.

Apparently press is getting interested in the whole affair. Richard Monastersky from the Chronicle of Higher Education in Washington D.C., was on the phone to Daniel earlier today and was trying to discover as many details possible. Certainly, once he discovers that there is no hoax - or at least, if there is, the rotten fish lands on the doorstep of MIT and Jackiw - he will abandon the subject. Perhaps it is true: MIT IS giving out Ph.D.s in "Science Fiction?" But, is that really fair?

Daniel has told us that:

I acted more like a notary, or a Journal editor: I understand the general idea, which makes some sense (even if I don't share it) and then relied on peer review.... That was certainly not the worst thesis in physics that has ever passed, and it got only a rarely given passing mark.

They are certainly sincere and desperately imprecise in their formulations, even in practical life (e.g. they could not give me the correct name of a mathematical Journal to which one of their papers is submitted). That is a kind of scientific dyslexia. But one should not pick on them for details, which will always be imprecise and even somewhat incorrect unless one rewrites the paper for them and prevents them from introducing last-minute brilliant looking and incorrect statements. Any good graduate student can do that in their stead, provided they (in particular Grichka) have a point and an original one.

The only important point is whether they really have an interesting point of view, or if this is only a fata morgana. Many distinguished scientists felt (at least, at first) that the former is correct. And it was easy for their enemies to resuscitate criticism because of the formal problems.

At a far different level, Moshe was somewhat like that in developing his ideas, because when saying and/or writing something his mind would already be several steps ahead and he would not pay attention to the details of what he says or writes; and then he would be angry if [anyone] would point to the formal errors instead of trying to understand what he meant.

We understand that the discussion following a standard Ph.D. defense lasts about half an hour. Apparently the discussion about the Bogdanov defense went on for over two hours, more or less deadlocked, until finally it was decided that nobody could put their finger on why they felt ill-at-ease about the matter.

As Ark is fond of saying: The Devil is in the details.

One of the recently revealed details in this matter is that the Bogdanovs have raised the ire of science by crossing over into metaphysics. We notice that the letter from the permanent member of Laboratoire Gevrey de Mathématique Physique, quoted at the beginning, describes the Bogdanov work as "impressionistic." We think that the very fact that they managed to convert "impressionistic" ideas into something that approaches hard science is a detail that deserves some scrutiny.

Ark has written about his concern over the fact that there has been no real progress in physics for over 50 years. Of course, we have the idea that this is possibly deliberate - a result of a sort of COINTELPRO infiltration of the various fields of science. Ark has written in his essays on Physics and the Mysterious:

Let me, first of all, share with you my views on the state of physics today. More on this subject can be found in my lecture Bioelectronics As Seen by a Theoretical Physicist. Even though this lecture was given at a bioelectronics symposium more than ten years ago, nothing really has changed since that time, and part of the predictions given there have already come true, so I am only repeating here much of what I said then. (I plan to post this entire lecture as soon as the translation is complete.)

Physics is what physicists do. And physicists do what they are paid to do. This is one of the reasons why so many of the brightest minds work on a short-time-scale reward basis, doing what is fashionable at a given time. This is the main reason why there is no progress at all in the fundamental areas. The clash between Einstein's relativity theories - which describe classical gravity at macro-scales, and Bohr-Dirac-Heisenberg-Schroedinger quantum theories, providing phenomenology of micro-phenomena, - this clash is today even more dark and scary than it was seventy years ago.

There is no real progress.

Quantum Theory is supposed to be the greatest invention in science since the beginning of the study of deeper realities. The greatest success of Quantum Theory is considered to be Quantum Field Theory, such as the theory of a quantized electromagnetic field (photons) in interaction with quantized charged matter (electrons). The problem is, this theory is mathematically inconsistent. It involves wishful thinking rather than rigorous science! The only quantum field theories (in four dimensional space-time) that ARE free of contradictions, are so-called trivial ones; that is theories that describe particles that do not interact at all. These theories are mathematical exercises involving particles that are "dead," that will never form atoms. It seems to be that a universe that is governed by quantum field theories that are free of contradictions would be a dead universe, a universe of no interaction.

One can build a non-trivial quantum field theory, which may even describe something real or interesting, but then it would necessarily contradict Einstein's Special Theory of Relativity; it would be a non-relativistic one.

This is the dilemma. If you want to have both Quantum Field Theory AND Einstein's Theory of Relativity, then you've got a problem.

Thus, nontrivial relativistic quantum field theories in four space-time dimensions are divergent - they lead to infinities, and are mathematically inconsistent. Searching for the cure in fancy formal math (supersymmetry, superstrings, quantum groups) just does not work. New - fundamentally new - ideas are needed. Quantum theory is not understood at all - everyone is trying to "cook" by "changing" the recipes to suit the ingredients they have on hand, and this very often results in "Rock Soup."

Part of the present-day problem is that Niels Bohr succeeded in molding the minds of so many theoretical physicists into the "no need to understand" mode and this has done a great disservice to the science, the new generations of scientists, and most of all, to humanity. In this day and time, it could be said, that we more desperately need to understand the Order of the Universe than ever before!

Yet there is hope. There are areas, even in the "recognized physics" where NEW is still possible. And, it is possible because more and more physicists understand how little they understand about quantum theory. Physicists are realizing, little by little, that even in such established areas as macroscopic electrodynamics there are problems that need major new rethinking: railguns, exploding wire arcs, sonoluminescence, present us with problems that are not easily answered within the standard paradigm and need, perhaps, a major re-thinking of the foundations.

Some of the problems are that we do not really understand the physics of conductivity and superconductivity. We realize that macroscopic quantum effects are more common than we ever thought. Sure, it is evident to everyone who goes to Circuit City that technology is progressing pretty fast in these areas; but the same cannot be said about our understanding!

What about gravitational physics?

Many of the important questions are still unanswered. The old Mach principle is still a subject of serious debate and we do not know what to do with singularities like black holes. They badly need quantum physics, but, once again, quantum physics becomes inconsistent when married with gravity. So we really do not know where we are.

We do not know if gravity is a fundamental force or, perhaps, it is a collective and composite phenomenon. Some physicists want to explain electromagnetism in terms of gravity. Others want to derive gravity from electromagnetism.

There is a lot of talk about antigravity or gravity shielding at the most fundamental levels and perhaps "antigravity" or gravity shielding is a real effect? No one can agree, and little progress is being made except to disagree. You would be amazed at the battles that rage in the ivory towers of academia!

We do not even know (at least not from textbooks or physics journals) if antimatter is attracted or repelled by matter. Perhaps tachyons - particles travelling faster than light - do exist? Perhaps space-time can have causal loops and telephoning into the past is possible? Perhaps quantum tunneling phenomena involves sending information faster than light? Perhaps magnetic monopoles exist and play an important role in biological systems? Or, perhaps, the fifth dimension is more than just a mathematical device of providing a unified description of gravity and electromagnetism?

Who knows?

All these topics ARE discussed in professional journals, but with no conclusion, no agreement, no cigar.

[...]

I am considered to be an "expert" but many of my views are not shared by other experts. I believe that my mind is more open than the minds of many of my colleagues. On the other hand, some of these colleagues believe that my mind is TOO open. So I have to hide from them many of my beliefs and not speak to them about a lot of things that I know. In this way I can publish papers in mainstream journals, speak at conferences, organize conferences and have a pretty good reputation. But to preserve this reputation I need to be very careful - just making a hint here and there that what I do publish is not all that I would like to tell....

I think I really need to tell you these things so that you will NOT get an erroneous idea that all physicists are of the same opinion. They are not. University physics is pretty conservative - which is not a bad attitude at all. We do need to be conservative - this distinguishes science from poetry and daydreaming. But, being too conservative has, in the past, been a great barrier and hindrance to scientific revolutions. If being conservative and "scientific" was the only correct approach, then we would have solved all the mysteries of our existence in the past several hundred years of the "age of science!" The truth is: we are only at the beginning.

But, perhaps I AM too open minded.... Perhaps my colleagues are right in being skeptical about anything that is not "established science." I try to keep an open mind about THAT, as well!

I am of the opinion that physics must be always based on mathematics - the only reliable tool and a truly universal language. Without math we can talk about many things - but we are just TALKING. It is not yet science! And even if I believe that the domain of physics needs to be essentially extended, that it has to connect with - or even embrace - biology and psychology - that it has to become much less "physical" - it does not mean it needs to become less precise!

But still, no amount of math can take the place of the right inspiration. The study of physics consists in peeling away the layers of the outside appearances of things to reveal their hidden nature and meaning, and very often this inner nature is so deep and hidden that only mathematics can describe it. But, if there is no inspiration as to what might be the objective of the search, the peeling away process might end up being rather like an onion - when the layers are all gone, there is no longer anything there!

The new physics needs to be based on math - to an even greater degree than the old physics. It will be a new math, sure, but it will a rigorous math - a math of equations and algorithms and probabilities - a nonlinear math of complex structures and of transitions between these structures. The math of today is difficult and abstract, and the math of tomorrow might be yet more difficult to grasp, even if our computers will be able to do more and more of the abstract work for us. On the other hand, the new math may be incredibly simple and elegant - this could be the reason it has eluded the understanding of physicists today - that the most abstract of ideas are concealed behind a veil of utter, simple logic. This is why professional training is so important: it gives us tools, it teaches us the rigor of abstract thinking, it teaches us the logic of proving assertions, and it shows us the limits and uncertainties of mathematics itself. As we know from Bertrand Russell and Kurt Goedel: math has its paradoxes too!

As we have noted, the subject of the Bogdanov's work just happens to fall within the several areas of Ark's expertise. Perhaps their work is moving in the direction of this theorized "new physics." But, as Ark has said, it still needs to be based on math - and a rigorous math at that!

So it is - providing Igor and Grichka are willing - we will begin the process of finding that devil - or angel - and in examining the details, we may come to that bridge between science and mysticism.

In order to avoid getting tangled in these details, I will utilize colored fonts for the discussion below. We will update it as it proceeds!

Grichka Bogdanov: "Quantum Fluctuations of the signature of space-time at Planck scale." Igor Bogdanov: "Topological state of space-time at scale zero." Ark: The idea of changing of signature of space-time, in particular of going through the phase of "degenerate vierbein" has been discussed in my 1982 paper "Vanishing Vierbein". In fact the very idea of changing space-time signature is usually attributed to Andrei Sakharov (ref. 45 in "EEQT A WAY OUT OF THE QUANTUM TRAP" ) KMS states (KMS stands for Kubo-Martin-Scwinger) of the algebra of observables in quantum field theory and in statistical mechanics have been first rigorously invetigated by H. Araki, R. Haag, N.M. Hugenholtz, E. Stormer, J.D. Wieringa and M. Winnink, see references in "ON SOME GROUPS OF AUTOMORPHISMS OF VON NEUMANN ALGEBRAS WITH CYCLIC AND SEPARATING VECTOR" and "ON THE SPECTRUM OF INTERNAL SYMMETRIES IN QUANTUM FIELD THEORY" and also Chapter II.2.5 in "Algebraic methods in Statistical mechanics and Quantum Field Theory" by G. Emch. Note: On October 31 Grichka Bogdanov added: Dear Ark, Just an additionnal word regarding your comments on Grichka's thesis. It is true that signature change is an old idea. As a matter of fact in my bibliography I have quoted in ref. 445 the Sacharov paper (1984). But I have also indicated in the introduction of my thesis that this idea is still more ancient : it has been proposed in the context of constructive theory of fields by J. Schwinger (ref. 454)for the first time in 1959, Nacano (ref. 408) in 1963 and then by C.Lanczos in the context of euclidean methods (ref 324/325) in 1963. It's only much later (in 1978) that S.Hawking decided to apply this idea in the context of quantum cosmology. Sacharov came after. Now the point is that none of these authors have never worked on the idea of a possible "fluctuation of the spacetime metric". I presume I was the first to introduce in 1991 this idea in the premisses of my thesis this idea of quantum fluctuation of the signature between 3,1 and 4,0. This fluctuation can been described on formal basis by the fiber bundle sigma already mentionned in previous mails. Best Grichka

Ark to Igor and Grichka:

Dear Igor and Grichka,

 I am reading your paper on topological field theory and it does not make sense to me. Will you be willing to answer my questions?

ark

Igor and Grichka to Ark:

We are extremely happy that you give us some of your time to get an clearer view of what we are aiming at. But which thesis are you reading first? Igor or Grichka's? Better to begin with Grichka's because all the mathematical foundations are there. Specifically the chap 3 on Quantization (q-deformation) of the signature. Also chap. 4 on KMS spacetime.

Concerning the papers in english added in the annexes of Igor's thesis, it would be much better to read the printed versions (because there are some misprints in the version that we put on the server). Those papers are before corrections. So it might be more appropriate to go to the Classical & Quantum grav. site

http://www.iop.org/EJ/S/UNREG/lASNI6gOLyatsXlDGRoZHQ/toc/0264- 9381/18/21

in order to import the PDF version of the paper from there.

The same with KMS State of Spacetime at the Planck Scale (Annals of Physics)

http://www.idealibrary.com/links/toc/aphy/296/1/0

 Thank you again for your time, help and attention in this affair. Keep us informed and as a specialist of KMS states (they are not very numerous) do not hesitate to put all the questions you wish.

Best to you, bien amicalement,

 Igor / Grichka

 PS By the way we just got a call from a journalist (he works in Washington but I did not get the name of the journal : something in education. Sounds interesting.

Ark to Igor / Grichka

OK So let us start with your joint paper.

You write:

"Topological field theory is usually defined as the quantization of zero, the lagrangian of the theory being either i)zero mode or ii) a characteristic class of a vectorial bundle V"

Questions:

1) what precisely vector bundle you have in mind? 2) How precisely you define "zero mode" of the bundle 3) How precisely you "quantize"

ark

Igor/Grichka to Ark:

  You write:    
 "Topological field theory is usually defined as the quantization of   zero, the lagrangian of the theory being either i)zero mode or ii)   a characteristic class of a vectorial bundle V"     

Questions:    

 1) what precisely vector bundle you have in mind?

In this case it is a general standard definition. For example one can find this definition in ref. 19 (Fre, Soriani). So in this preliminary context the vector bundle is general.

2) How precisely you define "zero mode" of the bundle

Same remark as above. But let's now go back to what cohomological topological field theory is. In the TFT context, the background metric maybe present; but the lagrangian (or hamiltonian) of the system is given by L = 0 or a topological invariant such as L = F exterior F dual (F being the field strengh (in our context it is the curvature R).

M. Kaku recalls that Witten originally created topological field theory in order to use the sigma model as a tool to construct topological invariants for manifolds. Independently, working in 4 dimensions, Donaldson startled the world of mathematics by creating new topological invariants using the input of physics. For example, the first Donaldson invariant (see the introduction of CQG paper).

Now : what is the interest of topological field theory in our case ? It means that when the lagrangian of the theory is 0 (or invariant) something relevant can still be said about the system. This is precisely the case in the context of the initial singularity. At this stage, we reach the 0 scale of spacetime (beta goes to 0 and g (coupling constant) goes to 0). Then - as we demontrated in chap. 5 of Grichka'as thesis- the lagrangian becomes : R exterior R dual (but to simplify let's say: R R dual). In other words, the lagrangian becomes an invariant (is reduced to the topological term R R dual).

To answer your question, genrally speaking, the 0 mode of the theory means L = 0. But in our case, it means that the SCALE of the system beta = 0.

   3) How precisely you "quantize"

Generally speaking, at the planck scale, what is in question is the quantization of gravitation. Of course this leads to tons of problems. So what we have done is to pick up just one element of the spacetime curvature (here : the metric tensor). Now : what happens if we want to quantize the metric tensor? (reasonable hypothesis at the planck scale).

Well, in our view (as opposed to what is done in the context of string theory) one must also "quantize" the signature of the metric itself.

And that's what we have done. By quantization of the signature, we mean q-deformation of the signature of the spacetime metric in the framework of quantum groups. For example, we have demonstrated that starting from the lorentz group SO (3,1), there is a canonical (and unique) path of q-deformation of the signature of the metric between lorentzian SO q(3,1) and euclidean SO q (4) metrics (or between Uq (so 3,1)) and Uq (so 4)) in terms of enveloping algebras). More precisely, we have shown, using a general theorem that we have constructed, that Uq (so 3,1))and Uq (so 4)) are related by a cocycle bicrossproduct. For more details see chap. 3 of Grichka's thesis (and more specifically theorem 3.3.2)

We hope that we have clearly answered your questions. We wait for the next ones.

 Best

I/G

Ark to Igor/Grichka:

 You write:
        "Topological field theory is usually defined as the quantization of zero, the lagrangian of the theory being either i)zero mode or ii) a characteristic class of a vectorial bundle V"  

Questions:  

1) what precisely vector bundle you have in mind?    

 In this case it is a general standard definition. For example one can   find this definition in ref. 19 (Fre, Soriani). So in this preliminary   context the vector bundle is general.

You did not answer my question. I know what vector bundle is. But: Which particular vector bundle you have in mind in your model?

     2) How precisely you define "zero mode" of the bundle     

Same remark as above.

The remark above does not define "zero mode". Can you please define it to me.

  But let's now go back to what cohomological   topological field theory is. In the TFT context, the background   metric maybe present; but the lagrangian (or hamiltonian) of the   system is given by L = 0 or a topological invariant such as L = F   exterior F dual (F being the field strengh (in our context it is the   curvature R).

 I am not asking this question. I would like you to define for me the "zero mode" of your vector bundle. But do please answer question 1) first.

  M. Kaku recalls that Witten originally created topological field   theory in order to use the sigma model as a tool to construct   topological invariants for manifolds. Independently, working in 4   dimensions, Donaldson startled the world of mathematics by creating   new topological invariants using the input of physics. For example,   the first Donaldson invariant (see the introduction of CQG paper).

I am not asking about this. I would like you to define for me the "zero mode" of your vector bundle. But do please answer question 1) first.

   Now : what is the interest of topological field theory in our case ?   It means that when the lagrangian of the theory is 0 (or invariant)   something relevant can still be said about the system. This is   precisely the case in the context of the initial singularity. At this   stage, we reach the 0 scale of spacetime (beta goes to 0 and g   (coupling constant) goes to 0). Then - as we demontrated in chap. 5   of Grichka'as thesis- the lagrangian becomes : R exterior R   dual (but to simplify let's say: R R dual). In other words, the   lagrangian becomes an invariant (is reduced to the topological term R   R dual).

Again: I am not asking about this. I would like you to define for me the "zero mode" of your vector bundle. But do please answer question 1) first.

  To answer your question, genrally speaking, the 0 mode of the theory   means L = 0. But in our case, it means that the SCALE of the system   beta = 0.

Sorry, but L is not defined. You would have to define L first.

    3) Precisely how you "quantize"     

Generally speaking, at the planck scale, what is in question is the   quantization of gravitation. Of course this leads to tons of problems.   So what we have done is to pick up just one element of the spacetime   curvature (here : the metric tensor). Now : what happens if we want   to quantize the metric tensor? (reasonable hypothesis at the planck   scale).

This is not answer to my question. I want to know precisely how you quantize.     

We hope that we have clearly answered your questions. We wait for the   next ones.

 In fact you did not. I hope you will :-)

ark

Dear Lark,

We came back too late yesterday night to answer your questions. Here we
are, trying to meet your interrogations.

Questions:

1) what precisely vector bundle you have in mind?

The vector bundle we have in mind in our approach (superposition of
lorentzian and euclidean signatures) has nothing to do with the one involved in this general definition as referenced. In our construction of a bundle "sigma", the total space is : SO(3,1) x SO (4), the base SO(3,1) x SO (4) over SO (3) and the fibre is SO (3).

Starting from this, we have constructed the topological quotient space R 3,1 x R 4 over SO (3). (See chap 1 and 2 of Grichka's thesis).

2) How precisely you define "zero mode" of the bundle

The "zero mode" corresponds to the origin of the topological space associated to the vector bundle (see page 8 and 9 of Grichka's thesis).

To answer your question, genrally speaking, the 0 mode of the theory
means L = 0. But in our case, it means that the SCALE of the system beta = 0.

Sorry, but L is not defined. You would have define L first.

In our case (superposition of lorentzian and euclidean signatures) the lagrangian L at the planck scale is of the general form :

L = beta R + 1 over g squared x R squared + alpha R R dual. Here R is the Einstein term, R squared the higher derivative associated to the curvature in N=2 supergravity, and R R dual, the topological term. 1 over g squared is the dilaton and alpha the axionic superpartner of the dilaton.
3) How precisely you "quantize"

This is not answer to my question. I want to know how precisely you quantize.

In chap 3 of the thesis we detail our approach concerning quantization. We quantize the metric. This quantization is based on our crossproduct construction.

We have to run now (a seminar) But we'll come back with more detailed answers if you wish,

Best and thank you again for your kind attention,

Igor / Grichka

Ark to Igor/Grichka:

On 30 Oct 2002 at 13:53, igor.bogdanov wrote:

In our
construction of a bundle "sigma", the total space is : SO(3,1) x SO (4), the base SO(3,1) x SO (4) over SO (3) and the fibre is SO (3).

Question: which SO(3)?

The "zero mode" corresponds to the origin of the topological space associated to the vector bundle (see page 8 and 9 of Grichka's thesis).

Please DEFINE zero mode of your bundle. But please define your bundle
first. Which SO(3)?

In our case (superposition of lorentzian and euclidean signatures) the lagrangian L at the planck scale is of the general form :

L = beta R + 1 over g squared x R squared + alpha R R dual. Here R is the Einstein term

Einstein term of WHAT? What R has to do with your bundle?

ark

Igor/Grichka to Ark:

In our construction of a bundle "sigma", the total space is : SO(3,1) x SO (4), the base SO(3,1) x SO (4) over SO (3) and the fibre is SO (3).

Question: which SO(3)?

Let the elements g1 belonging to SO(3,1), g2 belonging to SO(4), and h belonging to SO(3). SO(3) - being the maximal compact subgroup of SO(3,1) and SO(4) - is diagonally embedded in SO(3,1) x SO(4).

( This embedding is characterized by a left action of SO(3) on SO(3,1) x SO(4). Then the couple (g1, g2,) is identified to (g1, g2,)---> (h g1, g2 h -1). The above axion defines principle bundle sigma, the structural group being SO(3) ).

---

Please DEFINE zero mode of your bundle.

Let's consider the partition function describing the states of the metric (in terms of signature). The "superposition" state of the signature of the metric is described by the quotient topological space "sigma top" constructed from the bundle sigma.

What we call the zero mode here corresponds to beta= 0 (scale 0) in the partition function of the metric states. This "0 mode" corresponds also to the origin of "sigma top".

---

In our case (superposition of lorentzian and euclidean signatures) the lagrangian L at the planck scale is of the general form :

L = beta R + 1 over g squared x R squared + alpha R R dual. Here R is the Einstein term

Einstein term of WHAT?

Of the "extended" gravitational theory (by "extended we mean including the squared term R2 and the topological term R R dual).

What R has to do with your bundle?
R is the classical component of the lagrangien L .

Remember that our lagrangien is a quantum lagrangien mixing the ultraviolet (topological) and the infrared (classical) sectors of the "extended quatum" theory. For beta goes to infinite, we reach the infrared sector of the theory, which means that the bundle sigma is then projected to SO(3,1) over SO(3) -lorentzian metric-. On the
contrary, on the ultraviolet limit, the only term that contributes to the lagrangian the topological term R R dual. In this case, the bundle sigma is projected onto SO(4) over SO(3) -euclidean metric-. This last case corresponds to what we call the "0 mode" of the beta in the partition function of the states of the metric.

Hope we were clear enough.

Amicalement,

I/G

Ark to Igor/Grichka:

Let the elements g1 belonging to SO(3,1), g2 belonging to SO(4), and h belonging to SO(3). SO(3) - being the maximal compact subgroup of SO(3,1) and SO(4) - is diagonally embedded in SO(3,1) x SO(4).

Probably what you mean here is:

SO(3) - being the maximal compact subgroup of SO(3,1) - is diagonally embedded in SO(3,1) x SO(4). that is "and SO(4)" was not intended.

( This embedding is characterized by a left action of SO(3) on SO(3,1) x SO(4). Then the couple (g1, g2,) is identified to (g1, g2,)---> (h g1, g2 h -1). The above axion defines principle bundle sigma, the structural group being SO(3) ).

Well, probably you mean something else. The embedding is not characterized by a left action. It is left action that is defined by the embedding.

But now we have a problem. Please come back to my original question:

1) what precisely vector bundle you have in mind?

Now, you wrote: "The above axion defines principle bundle sigma,"

But in your paper you write "vector bundle", not "principal bundle."

So, my question remains:

1) what precisely vector bundle you have in mind?

ark

I/G to Ark:

Ark wrote:
I would appreciate if you could send me some of your texts that are available as computer text files. I found one such thing: http://www.spiritual-dimension.com/dieu&science.html
but I would appreciate if you help me to find more on the net or as files that you can send me.
This will help me to have a better picture of the reality!

Dear Ark,

The problem is that there is no text files about Dieu et la Science available on the net. Since in was a great success in europe (the book was translated in 35 different countries except USA), there are numerous comments and websites about it. None of them are of our own (and most of them are not very relevant).

About the book : it was based on a dialogue with the big french philosopher Jean Guitton (personal friend of the Pope Paul VI and main initiator of the Council Vatican II). In this dialogue we have tried to explore the "rational theology" derived from Saint Thomas d'Aquinas.

In fact, you have put the finger on the main (and in fact the unique) source of all our troubles. Because, considering 1) the subject (God and Science) and 2) the enormous success of this book (beyond 1 million copies for France only), the whole science community fell on us. We had articles and papers raised by the "Union Rationaliste" claiming that the encounter between science and metaphyisics was a sort of a conceptual crime.

But if you read the book you wil see that as far as we represent science we were always very cautious and extremely conservative and prudent.

If you wish, we can send you a french version of the book (In fact we will do the same for Jackiw who asked us to mail him a signed copy of this book)

Best,

I/G

Ark to I/G:

So, here we have a problem. A real problem. I asked "which vector bundle?" You answered "SO(3,1)xSO(4)/SO(3)"

But this is NOT a vector bundle. So, now, you are changing your story.

You say: "the vector bundle V which is involved in this definition is the one from which Donaldson invariants are obtained."

Yet this does not answer my question.

I am therefore asking again: "Which vector bundle?" Will you be so kind as to answer in as precise way way as you can, this one question? WHICH vector bundle? The bundle you have suggested is NOT a vector bundle, so I really wonder why you would even mention it? But we can leave it till later. Let us stay with the original question:

1) Which vector bundle?

Let us remember that a vector bundle can be defined in several ways. So ANY one will do. But we do need ONE at least.

No answer received as of October 30, 2002. Ark writes again:

Dear Igor,

My question 1) is not yet answered.

Which vector bundle?

First you tried to answer by defining the total space, the fiber, the basis (homogeneous space), but I was asking about VECTOR bundle. Again we need the base, the fibre, and some way of defining the global stucture. Only when we have this, we can try to define some L. But we will come to L, R, F etc. later on. First you need to define the arena.

So, which exactly vector bundle you discuss?

ark

Dear Igor,

When you are tired with answering my "interrogation" (as you have described it) about vector bundles for a change we can move to factors, von Neumann algebras etc.

Here my first question would be in the direction of understanding the terminology and notation that you are using.

On p.83 of your thesis document you write:

"There exist three types of factors: the type I and type II which are commutative...."

Now, please, help me here: a factor has a trivial centre. A factor that is commutative would be trivial. Therefore the sentence as it stands makes no sense. Would you agree?

ark

I/G to Ark:

"There exist three types of factors: the type I and type II which are commutative...."

Now, please, help me here: a factor has a trivial centre. A factor that is commutative would be trivial. Therefore the sentence as it stands makes no sense. Would you agree?

Very good point! Yes of course we agree. But not in the limits of our own approach. The type I and II factors we use here are (by definition) endowed with a trace. In fact those factors are of course not commutative algebraically speaking. But in our approach we consider a property of "commutativity under the trace".

By "commutativity under the trace" we mean, as usual : tau (A B) = tau (B A).

This notion of "commutativity under the trace" came from professor F. C. (one of the main founders of the "weight theory" in the early seventies). We had so many discussions with him employing this notion that it became quite natural to us. But yes : the type I and II factors are NOT commutative and in the paper, by misprint, the additive precision "under the trace" was not present. In the last printed KMS paper (Chin.Jour. of Phys.paper that was printed after) this terminology problem was fixed.

But yes, the statement about the commutativity of type I, and II is - stritcly speaking inapropriate. (Examples: Matrix algebras, Fermi fields admit tracial states giving rise to type I resp. II representations).

In our case, this condition of commutativity under the trace applies to the specific factors I and II infinite involved in our approach.

In particular, considering the Conne's group

Out of M = Aut of M / Int of M

we see this completly canonical group as the sole manifestation of the non commutativity of the albebra M (which in this case is, of course, a III lambda factor). Obviously, the KMS states are not really relevant for type I and II. One can consider these I and II cases as giving "empty KMS states". The reason? because in these two cases (tracial states) the modular group of the system is trivial; as a result, all the modular automorphisms are "inner automorphisms" which is supressing all relevant contents of Tomita's theory. An other way to see it is the following :

considering the Conne's group :

Out (M) = Aut(M)/ Int(M)

in the trivial I and II cases we are left with

Aut (M) = Int (M)

because the modular group is trivial and the flow of weight givent by Aut of M is reduced to identity.

Best to you, and thank you again for your help,

Igor / Grichka

Ark to I/G:

But yes : the type I and II factors are NOT commutative and in the paper, by misprint, the additive precision "under the trace" was not present

OK So it was missprint. Probably another missprint is where you write that type III is "traceless". In mathematics the term "traceless" usually means "of trace zero." Now, I want to understand the next
sentence. You write:

A trace tau on a factor M is a linear form such that tau(AB)=tau(BA) for all A,B in M.

Which is followed by: "In this case any measure on M is invariant."

My questions:

1) What do you mean by "In this case"? "This?" Which?
2) What do you mean by "measure on M"?
3) In case you answer 1 and 2 - what do you mean by "is invariant"? What is invariant and with respect to what?

ark

OK So it was missprint. Probably another missprint is where you
write that type III is "traceless".

I/G to Ark:

Answer : Yes. We write all our papers directly in english (which is not our native language). So we meant : "without a trace" (and not of trace zero).

In mathematics the term "traceless" usually means "of trace zero."

Now, I want to understand the next sentence. You write:

" A trace tau on a factor M is a linear form such that tau(AB)=tau(BA) for all A,B in M. Which is followed by: "In this case any measure on M is invariant."

My questions:

1) What do you mean by "In this case"? "This?" Which?

Answer : We mean when the factor M has a trace

2) What do you mean by "measure on M"?

Answer : We mean "measure on W (M)" . Where W (M) is an ordinary measurable space endowed with a measure class. We consider on W (M) the 1 parameter group (W k) k belonging to R which is a flow whose natural parameter is R dual, dual of R. This flow has "intrinsic" properties in terms of class of weights on M and is the "flow of weights of M". Or course, in the case of factors of type I and II, a weight phi on M is a
trace if phi is invariant under the inner automorphisms of M.

3) In case you answer 1 and 2 - what do you mean by "is invariant"? What is invariant and with respect to what?

Answer : That means that the weight phi on the von Neumann algebra M is a trace. The weight phi is invariant under (or by) the inner automorphisms of M.

Best

I/G

Ark to I/G:

On 31 Oct 2002 at 21:15, igor.bogdanov wrote:

2) What do you mean by "measure on M"?

Answer : We mean "measure on W (M)" . Where W (M) is an ordinary measurable space endowed with a measure class.

We have a problem here. What is "ordinary measurable space over M", when M is a von Neumann algebra? How you define it?

ark

I/G to Ark:

Answer : We have not written "ordinary measurable space over M" but :

W (M) is an ordinary measurable space endowed with a measure class

I/G

Ark to I/G:

How you define W (M)? Remember: M is a von Neumann algebra.

I/G to Ark:

Of course. M is a von Neumann algebra (can be factor I, II or III). Here W (M) is the flow of weight of the algebra M. More precisely, let M be a von Neumann algrebra. Then Connes as shown that there exists a
canonical homomorphism of the form : f : R --> Out (M) = Aut (M) / Int (M) - which does not depend on any auxilary choice. So the algebra M determines an intrinsic dynamic f which is a one parameter automorphism group (Aut (M) modulo the inner automorphis Int (M). This canonical dynamic f is trivial when M happens to be a factor I or II (in this case, the flow of weights becomes a trace). In the case of factor of type III, f allows to define an invariant of M which is an ergodic flow W (M), W (lambda) where W (M) is a measurable
space endowed with a measure class. W (lambda) - with lambda belongs to R* + - is a I parameter group of transformations (or flow on W (M)). The natural parameter of this group is R* + dual of R. As Alain Connes has demonstrate it, the action of R* + is simply the multiplication phi ---> lambda phi and is precisely the flow of weights of the algebra M.

Best

I/G

On 1 Nov 2002 at 16:06, igor.bogdanov wrote:

Of course. M is a von Neumann algebra (can be factor I, II or III). Here W (M) is the flow of weight of the algebra M.

Please, do answer my question:

You say

a) W (M) is a measurable space endowed with a measure class.
b) W (M) is the flow of weight of the algebra M.

Let us keep a) - your original attempted answer.
Please do define me W(M).
I know what a measurable space is and I know what measure class is - in general.
Now, please define W(M), where M is a von Neumann algebra. So that I understand what you are talking about. The devil - always - is in the details.

ark

I/G to Ark:

Dear Ark,

Before going back to your questions, I thought it would be necessary to clarify an other rumor regarding the so called reaction of Classical and Quantum Gravity after the publication of our paper.

This rumor suggests that Class.&Quant.Gravi might have decided not to work anymore with the physicist who reviewed our paper (prentending that the referee did not even read the paper).

We do not think that such a rumor is grounded. If it was the case, the journal's decision would be a regrettable mistake.

In order to bring tangible elements to the debate, we send you hereafter all the reports that were induced by the submission of our paper in Classical and Quantum Gravity.

As you will see, it took 7 month of exchanges and very hard work before the paper could be accepted in this high standards journal.

Subject:
CQG/119461/PAP
Date: Thu, 15 Feb 2001 10:45:22 +0000
From: "Imported For: Class. Quantum Grav. --By: IOPP.API" <cqg@iop.org>
To: <igor.bogdanov@free.fr>

Ref: CQG/119461/PAP

15 February 2001

Dr I Bogdanov
Laboratoire Gevrey de Mathematique
Physique
Universite de Bourgogne
CNRS UPRES A 5029
5 avenue de Montespan
75016 Paris
FRANCE

Dear Dr Bogdanov

TITLE: Topological theory of the initial ...
AUTHORS: G Bogdanov et al

We have now received advice from our referees on this Paper, which is under consideration for Classical and Quantum Gravity, and are enclosing copies of the relevant report(s) recommending substantial
changes.

First Referee's Report

According to the cover page of the manuscript sent to this referee, this paper was received electronically. I believe there are several typographical errors that may have be due to the transmission or
software problems (e.g., possibly some extra h's). Thus, if this paper is accepted, then the authors and editors may wish to be extra careful in going over the printed proof copy.

Page 7 line 4: There is an extra "h" after the sentence at the end of the first paragraph.This may be a transmission or software error.

Page 9 line 3: There is an extra "h" after the sentence which ends with "a topological invariant."

Page 12 line 7: "filed" (after the word quantum) should be "field".

Page 12 line 10: "symmetric" (after Euclidean) should be "signature".

Page 20 line at bottom: There is an extra "h" after the last sentenceon the page.

Pages 22-24 (References): The references on the manuscript sent to this referee are not quite in the usual style of the journal. Also, they are not even given in a consistent style. In particular, some multiple authored articles have a comma between the authors names (see for example reference #5) and some multiple authored papers do not have a comma between the authors names (see for example reference #7). For unknown reasons the references #11, #12 and #14 have parens around the letter of the first name of the author.

Second Referee's Report

QUALITY ASSESSMENT: Q2, Sound, original and of interest.

RECOMMENDATION: R4: Revise substantially, along the lines indicated;
with these revisions I expect the paper to be suitable for
publication.

The author's make the interesting observation that, in the limit of infinite temperature, a field theory is reduced to a topological field theory which may be a suitable description of the initial phase of the universe. I recommend the following points be clarified in the paper before publication:

(1) Through out the paper, \beta = 0 is stated and it would be much clearer if \beta -> 0 is considered which better describes the limit of infinite temperature.

(2) On page 4 (and other pages e.g. page 6) \beta -> \dot which should presumable be replaced by \beta -> \infty? There is also a reference missing on page 4.

(3) Much of the details in section 2, regarding the metric independence of the partition function, are standard details which could be omitted. Also, the form of the energy - momentum tensor T_{\alpha\beta}, given on page 8, is true for a specific type of field theory. The authors' provide no information of the nature of field theories being considered in the paper. For example, are they supersymmetric etc.?

(4) The authors' point out the H=0 (or L, which is typical for topological field theories) can, more or less, be viewed as the same as \beta H =0 for \beta =0 (in the limit of infinite temperature).

This crucial and interesting observation needs to be supplemented with more detailed analysis since it is crucial for their ideas to work. It would be very helpful and more convincing if the authors' could provide further support. For example, can contact be made with general covariance or topology on taking the \beta->0 limit of some established standard results?

(5) I can almost accept that in the limit of infinite temperature, contact can be made with a topological phase of some field theory (the type of field theory needs to be elaborated on however). The crucial question, however, is how does the initial topological phasebreak down to a universe we see today. I would be of great interest if the authors' could at least worry about this issue.

(6) The scale of metric mentioned in proposition 2.2 is not easy to understand.

(7) In some places, the grammar used needs to be re-worded. Also, the various "black dots" which appear throughout the paper are confusing and need clarifying.

If the author's can successfully rectify the above, I will recommend the paper for publication.

Subject: CQG/119461/PAP
Date: Mon, 23 Jul 2001 14:52:13 +0100
From: "Imported For: Class. Quantum Grav. --By: IOPP.API" <cqg@iop.org>
To: <igor.bogdanov@free.fr>

COMMENT : On the basis of this very detailed set of questions / corrections and requests of modifications 5 month of work was necessary to meet all the requests of the referees. We sent the revised version mid June and received the second report on July 23 as follows :

Ref: CQG/119461/PAP

23 July 2001

Dr I Bogdanov
Laboratoire Gevrey de Mathematique
Physique
Universite de Bourgogne
CNRS UPRES A 5029
5 avenue de Montespan
75016 Paris
FRANCE

Dear Dr Bogdanov

TITLE: Topological theory of the initial ...
AUTHORS: G Bogdanov et al

We have now received advice from our referees on this Paper, which is under consideration for Classical and Quantum Gravity, and are enclosing copies of the relevant report(s).
--

Second Referee's Second Report

The revised manuscript is much better but still requires some re-working on the grammar, which I will not make an issue of.

In a few places, however, the authors need to make it clear what they are referring to, for example on page 33, the line after equation (77) states: "Then, as showed in (2.1), ....". Are the authors referring to example (2.1)? I will assume such issues will be rectified.

Regarding content, the authors have addressed my original questions

(1) Through out the paper, \beta = 0 is stated and it would be much clearer if \beta -> 0 is considered which better describes the limit of infinite temperature.

(2) On page 4 (and other pages e.g. page 6) \beta -> \dot which should presumable be replaced by \beta -> \infty? There is also a
reference missing on page 4.

(3) Much of the details in section 2, regarding the metric independence of the partition function, are standard details which could be omitted. Also, the form of the energy - momentum tensor T_{\alpha\beta}, given on page 8, is true for a specific type of field theory. The authors' provide no information of the nature of field theories being considered in the paper. For example, are they supersymmetric etc.?

(4) The authors' point out the H=0 (or L, which is typical for topological field theories) can, more or less, be viewed as the same as \beta H =0 for \beta =0 (in the limit of infinite temperature).

This crucial and interesting observation needs to be supplemented with more detailed analysis since it is crucial for their ideas to work. It would be very helpful and more convincing if the authors' could provide further support. For example, can contact be made with general covariance or topology on taking the \beta->0 limit of some established standard results?

(5) I can almost accept that in the limit of infinite temperature, contact can be made with a topological phase of some field theory (the type of field theory needs to be elaborated on however). The crucial question, however, is how does the initial topological phasebreak down to a universe we see today. I would be of great interest if the authors' could at least worry about this issue.

(6) The scale of metric mentioned in proposition 2.2 is not easy to understand.

(7) In some places, the grammar used needs to be re-worded. Also, the various "black dots" which appear throughout the paper are confusing and need clarifying.

and I feel that by doing so (authors) have needed to include a lot more detail than they originally intended, but I feel the paper may be made more accessible if the following was considered:

Section 5 is very important and interesting but I feel it can be simplified. Is it possible to provide a specific example or toy-model of the ideas here? Then, perhaps, an outline of the detailed results could be given? The authors point out that there are further details in ref. 2, and I feel that many of the details of section 5 could be left their, but this may not be possible?

COMMENT : We worked again during a whole month before sending the new substantially revised version to CQG (Aug 15). Then we received the accetance of the paper on Aug 24 :

Subject: CQG/119461/PAP
Date: Fri, 24 Aug 2001 15:19:56 +0100
From: "Imported For: Class. Quantum Grav. --By: IOPP.API" <cqg@iop.org>
To: <igor.bogdanov@free.fr>

Ref: CQG/119461/PAP

24 August 2001

Dr I Bogdanov
Laboratoire Gevrey de Mathematique
Physique
Universite de Bourgogne
CNRS UPRES A 5029
5 avenue de Montespan
75016 Paris
FRANCE

Dear Dr Bogdanov

TITLE: Topological theory of the initial ...
AUTHORS: G Bogdanov et al

We are pleased to inform you that we have accepted your article for publication in Classical and Quantum Gravity as a Paper.

COMMENT : It took 7 month of very detailed work before we could address all the questions raised in the various reports and get a version of the paper that could satisfy the referee.

After having read the referee's report, everyone who is familiar with topological field theory would immediately realize that the referee understood perfectly well the paper. His question raised in point 4 demontrates that not only the referee obviously knows the principles of topological field theory but also understood the idea presented in the paper : " The authors' point out the H=0 (or L, which is typical for topological field theories) can, more or less, be viewed as the same as \beta H =0 for \beta =0 (in the limit of infinite temperature)."

IN CONCLUSION : It seems totally unfair to pretend :

1) that the referee did not read the paper
2) that the referee did not understand its content
3) that the referee was not demanding profound corrections and modifications for
the paper to meet the standards of the journal.

Could you put this mail on your site in order to help your readers to
have a clearer point of view on this point?

Thank you for your attention,

Best regards,

Igor BOGDANOFF Grichka BOGDANOFF

Ark to I/G:

Thanks. But I am still expecting my questions.

1) Vector bundle
2) W(M)

We got e serious problem there.

ark

I/G to Ark:

Subject: Re: REPORT CLASS. 1 QUANTUM GRAVITY/BOGDANOFF
Date sent: Sat, 02 Nov 2002 00:52:26 +0100

Dear Ark,

I know exactly how to address your question. But we will do it tomorrow
(Helas, I must go terribly late).

Best

I/G to Ark:

Date sent: Sat, 02 Nov 2002 19:41:35 +0100

Answer : To be specific concerning W(M) : it is the flow of weight of the algebra M. The flow of weight W(M) represents the restriction to the centre Z(N) of the action of theta index lambda, lambda belonging to R*.

In our case, we have three types of flows corresponding to the three types of factors involved :

1. For beta > l/planck (classical scale) W(M) is a trace and the lebesgue measure on the metric is fully defined.

Consequently, the von Neumann algebra associated to the classical scale is a "Type I infinite" factor.

In this case W(M) = Tr(M).

2. For beta = O (topological scale) W(M) = Tr infinite (M). As all the measures performed on the euclidean metric are ro equivalent up to infinity, the system is ergodic. Alain Connes has shown that any ergodic flow for an invariant measure in the Lebesgue measure class gives a unique Type II infinite factor. This suggests that the singular 0 scale should be define by a Type II infinite factor endowed with a trace.

3. For beta between 0 and l/planck, we reach the KMS domain. Because of the quantum fluctuations of the metric the measure on this (non commutative) metric is ill defined. Therefore, M should be a traceless algebra of type III lambda.

In this case W(M) is : exp - beta h x M x exp + beta h

Note that in the Type I and Type II factor cases, the flow of weight W(M) is simply the "flow of translations" of R onto R. In the Type III case, the flow of weight can be understood as the action of R* over M (because of the III lambda decomposition into the semi direct product between a Type II infinite and R)

Best,

I/G

Ark to I/G:

On 2 Nov 2002 at 19:41, igor.bogdanov wrote:

Answer : To be specific concerning W(M) : it is the flow of weight of the algebra M.

If now you decide that W(M) is NOT "ordinary emasurable space" as stated before, but that W(M) is now a "flow of weight" - then we are back to square one. You did not answer my question. Let me, therefore, repeat my question of October 31 again:

"Which is followed by: "In this case any measure on M is invariant."

My questions:

1) What do you mean by "In this case"? "This?" Which?
2) What do you mean by "measure on M"?
3) In case you answer 1 and 2 - what do you mean by "is invariant"?
What is invariant and with respect to what?

ark

I/G to Ark:

What? your read W (M) for the flow of weights? This is very strange. Because we wrote W index lambda (M). But the conclusion is that the "index lambda" was suppressed by the mail transmission protocol (which
does not take these symbols).

So we understand now that there is a misunderstanding (apparently since the beginning).

Let's go back to the definitions.

(W (M), W index lambda (M)) is an ergodic flow where W (M) is a measurable space endowed with a measure class and W index lambda (M) is a one parameter group of transformations (or flow) on W (M). The parameter of this transformation is R*, dual of R. This flow is the flow of weights of M.

Can you confirm that you got the difference between W (M) (measurable space) and W index lambda (M) (flow of weights of M) ?

I/G

Ark to I/G:

On 3 Nov 2002 at 1:20, igor.bogdanov wrote:

Can you confirm that you got the difference between W (M) (measurable space) and W index lambda (M) (flow of weights of M) ?

I confirm. My questions 2 and 3 of October 31 remain unanswered. Should I conclude that you can't answer these questions and simply move forward?

ark

Ark to I/G:

Date sent: Sun, 03 Nov 2002 10:24:22 -0500

Dear Igor,

I have a comment on an earlier statement.

The section on KMS (p. 83 of your thesis) starts with

"In the KMS state, the only von Neumann algebras involved are what is called factors."

This statement makes no sense.

1) There is no such thing as "KMS" state.

There always is "KMS state of ... with respect to ...", where "of" is "of some algebra" and "with respect to" is "with respect to some 1-parameter automorphism group."

Sometimes "with respect to" is not expliciltly listed - when it is clear from the context which one-parameter group of automorphisms is considered.

2) KMS states of algebraic dynamical systems can be defined independently of whether the algebra is a factor or not.

I wonder how your thesis could ever go to through the referees! Could you tell me who were the referees responsible for the C*- algebraic part?

ark

[Ark's Note: Let me mention that in the process of my "critical examination" of Igor and Grichka physics publications I myself can make a mistake or an error. That happens. If somebody will ever find such an error, please let me know, and I will correct it asap and will made the correction known to the readers of these pages. A friend wrote to me today (Nov. 3) something worth of quoting:

Should I quote the names of several people whom I know (and whom you know as well) who do not know that there exist several inequivalent principal bundles with the same base, same total fiber, and same basis...or who do not make the difference between a principal bundle and a vector bundle (the worst of it is that some of these people are known as "specialists of geometry in physics" in the community of mathematical physicists !) Yes, this can be irritating, but in the case of the Bogdanov brothers, they look like if they were humble enough to ask your opinion and critics (others do not even care about their own lack of precision or even, competence).

]

Date sent: Tue, 05 Nov 2002 13:56:40 -0500
Dear I/GOur pages have been updated. Note in particular the bottom of the

http://www.cassiopaea.org/cass/bogdanovs.htm

page.

Yet I am still waiting for your answers to my questions. Are you SO busy that you can give short answers
to simple questions?

Best wishes,

ark

I/G to Ark:

Dear Ark,

Glad to be back to physics with you. As a matter of fact - as you know - we were taken by a huge storm during last 48 hours. We have had no respite untill now (almost 3 in the morning) to respond to your mails.
Before we address your questions regarding operators algebras, let us first express our greatfull feelings for your work on your site. It seems it has become the most important one regarding this affair and, beyond, it allows to decode all the hidden process that the scientific community does not seem to want to acknowledge. We are also thankfull to you for having published Dr. Coquereaux's letter which greatly helps everyone to focus on what is really happening in the scientific community.

But here are our answers to your last questions.

You wrote:

Dear Igor,
I have a comment on an earlier statement.
The section on KMS (p. 83 of your thesis) starts with
"In the KMS state, the only von Neumann algebras involved are what is
called factors."
This statement makes no sense.

1) There is no such thing as "KMS" state.

There always is "KMS state of ... with respect to ...", where "of" is "of some algebra" and "with respect to" is "with respect to some 1-parameter automorphism group."

Comment : We meant here KMS state in general (it is the reason why it
is not specified). In fact we purposly omitted to write the factor III
lambda factor involved.

Sometimes "with respect to" is not expliciltly listed - when it is clear from the context which one-parameter group of automorphisms is considered.

Comment : it was in respect to the automorphism group given by eq.51.

2) KMS states of algebraic dynamical systems can be defined independently of whether the algebra is a factor or not.

Comment : OK. We do not desagree with this. We should have said (but it
was implicit because of the reference to Grichka's thesis where it is
explained) : "In our model, KMS states..."etc. Because we have
restricted the general von Neumann algebras to factors.

I wonder how your thesis could ever go to through the referees! Could you tell me who were the referees responsible for the C*- algebraic part?

Comment : The C*algebraic work was developed in common with Grichka. All the basis of von Neumann algebras, KMS states, modular theory, weight theory, etc have been very carefully followed and checked line by line by E.L. (at that time at E.N.S.) who issued (January 2000) a written report to Daniel Sternheimer. Moreover the main aspects of von Neumann algebras etc. have been developped in the curse of numerous exchanges and fruitfull discussions with F.C. (University of Orleans). As you know, F.C. is one of the main founders of the theory of weights. Last, the operators algebras part has also been worked out in details with M.E. (Paris VII) and also discussed with P.C. In conclusion : no doubt that the reading and checking process of our work was very serious an acurate (because of the specific "Bogdanoff Effect" that was already playing against us at the time). >

We will try to be more specific in our next mail,

Best,

I/G

[Note: Names replaced by initials]

On 6 Nov 2002 at 2:40, igor.bogdanov wrote:

> > 1) There is no such thing as "KMS" state.
> >
> > There always is "KMS state of ... with respect to ...",
> > where "of" is "of some algebra" and "with respect to"
> > is "with respect to some 1-parameter automorphism
> > group."
>
> Comment : We meant here KMS state in general (it is the reason why it
> is not specified). In fact we purposly omitted to write the factor
> III lambda factor involved.

Sorry, but there is no such a thing as "KMS state in general."
If you think that there is such a thing - please define it.

> > Sometimes "with respect to" is not
> expliciltly listed - when it is > clear from the context which
> one-parameter group of automorphisms is > considered.
>
> Comment : it was in respect to the automorphism group given by eq.51.

Which particular automorphism group and of which algebra? You need to
specify the algebra and the automorphism group. Or you should say: we
mean ANY algebra and ANY aotomorphism group. One of the above.

> > > 2) KMS states of algebraic dynamical systems can be defined >
> independently of whether the algebra is a factor or not.
>
> Comment : OK. We do not desagree with this. We should have said (but
> it was implicit because of the reference to Grichka's thesis where it
> is explained) : "In our model, KMS states..."etc. Because we have
> restricted the general von Neumann algebras to factors.

Now you are saying something different from what you wrote. Good that
we are fixing these glitches. But a paper that has been published
should have most glitches fixed before publication.

> >
> > I wonder how your thesis could ever go to through the referees!
> > Could you tell me who were the referees responsible for the C*-
> > algebraic part?
>
> Comment : The C*algebraic work was developed in common with Grichka.
> All the basis of von Neumann algebras, KMS states, modular theory,
> weight theory, etc have been very carefully followed and checked line
> by line by Eric Leichtnam (at that time at E.N.S.) who issued (January
> 2000) a written report to Daniel Sternheimer.

As far as I can see the details have NOT been CAREFULLY checked.

> Moreover the main
> aspects of von Neumann algebras etc. have been developped in the curse
> of numerous exchanges and fruitfull discussions with F.C.
> (University of Orleans). As you know, F.C. is one of the main
> founders of the theory of weights.

Perhaps. But it you who are signed as the authors. And therefore full
responsibility is yours. I can believe that C. discussed with you
ideas, but I can not believe that C. checked your papers for
inconsistencies and errors.

> Last, the operators algebras part
> has also been worked out in details with M.E. (Paris VII) and
> also discussed with P.C. In conclusion : no doubt that the
> reading and checking process of our work was very serious an acurate

Perahps they were serious. But they were not accurate.

> (because of the specific "Bogdanoff Effect" that was already playing
> against us at the time). >
>
> We will try to be more specific in our next mail,
>
> Best,
>
> I/G

Yes, we need to fix the details. That is where devil resides.

ark

I/G to Ark:

Date sent: Wed, 06 Nov 2002 15:45:35 +0100

Arkadiusz Jadczyk wrote:
>
> On 6 Nov 2002 at 2:40, igor.bogdanov wrote:
>
> > > 1) There is no such thing as "KMS" state.
> > >
> > > There always is "KMS state of ... with respect to ...",
> > > where "of" is "of some algebra" and "with respect to"
> > > is "with respect to some 1-parameter automorphism
> > > group."
> >
> > Comment : We meant here KMS state in general (it is the reason why it
> > is not specified). In fact we purposly omitted to write the factor
> > III lambda factor involved.
>
> Sorry, but there is no such a thing as "KMS state in general."
> If you think that there is such a thing - please define it.

COMMENT : We agree. The problem might be that we assumed that the
context would help to understand that in the general reminder of KMS
condition we were refering to type I, II and III factors. Sorry if we
did not express the idea correctly.
>
> > > Sometimes "with respect to" is not
> > expliciltly listed - when it is > clear from the context which
> > one-parameter group of automorphisms is > considered.
> >
> > Comment : it was in respect to the automorphism group given by eq.51.
>
> Which particular automorphism group and of which algebra?

COMMENT : We do not mean here ANY algebra. We mean what we have called
the algebra M index q, which is a type III lambda factor. The 1
parameter automorphisms group is : sigma M = exp +iht x M x exp -iht.

>
> > > > 2) KMS states of algebraic dynamical systems can be defined >
> > independently of whether the algebra is a factor or not.
> >
> > Comment : OK. We do not desagree with this. We should have said (but
> > it was implicit because of the reference to Grichka's thesis where it
> > is explained) : "In our model, KMS states..."etc. Because we have
> > restricted the general von Neumann algebras to factors.
>
> Now you are saying something different from what you wrote. Good that
> we are fixing these glitches. But a paper that has been published
> should have most glitches fixed before publication.

COMMENT : absolutly. We fully agree with that.
>
> > >
> > > I wonder how your thesis could ever go to through the referees!
> > > Could you tell me who were the referees responsible for the C*-
> > > algebraic part?
> >
> > Comment : The C*algebraic work was developed in common with Grichka.
> > All the basis of von Neumann algebras, KMS states, modular theory,
> > weight theory, etc have been very carefully followed and checked line
> > by line by Eric Leichtnam (at that time at E.N.S.) who issued (January
> > 2000) a written report to Daniel Sternheimer.
>
> As far as I can see the details have NOT been CAREFULLY checked.

COMMENT : In Grichk'as thesis, yes. But not in certain parts of the papers.

> >
> Moreover the main
> > aspects of von Neumann algebras etc. have been
developped in the curse
> > of numerous exchanges and fruitfull discussions with
F.C
> > (University of Orleans). As you know, F.C. is one of the
main > > founders of the theory of weights.
>

Perhaps. But it you who are signed as the authors. And therefore full responsibility is yours. I can believe that C. discussed with you ideas, but I can not believe that C. checked your papers for inconsistencies and errors.

COMMENT : No, you are right.

> > Last, the operators algebras part
> > has also been worked out in details with M.E. (Paris VII) and
> > also discussed with P.C. In conclusion : no doubt that the
> > reading and checking process of our work was very serious an acurate

> Perahps they were serious. But they were not accurate.

COMMENT : In the PHD, as far as Leichnam is concerned, it is the case.
There is a written report on the operators algebra part.
>
> > (because of the specific "Bogdanoff Effect" that was already playing
> > against us at the time). >
> >
> > We will try to be more specific in our next mail,
> >
> > Best,
> >
> > I/G
>
> Yes, we need to fix the details. That is where devil resides.
>

Thank you, Best,

Igor / Grichka

Ark to I/G

Date: Wed, 06 Nov 2002 11:44:08 -0500

Dear I/G

Let us go back to my question no 2) of October 2.

2) What do you mean by "measure on M"?

ark

Date sent: Wed, 06 Nov 2002 14:46:58 -0500
Dear I/G

After some thinking, after reading what is being posted on the web,
how journalists distort the facts by selective and biased quotes,
I decided to change the formula of our discussion.
Till now I was essentially asking you to define precisely the terms
that you were using, EVEN when I could guess myself their meaning.
That is I was applying to your papers standards that are not common
in the physics. Applying these standards to other papers that has
been published in "respected" journals would cause rejection of at
least 80% of what has been published.
Such an approach would be fair if you were my students, educated
in an academic environment that makes clarity of expression an
important issue. But I realized that you had to simply grab the
knowledge from many different sources, not going through the
standard, high quality, line that produces "experts in the field".

Therefore my way of formulating questions will change. Instead of
trying to extract fom you things that I know, but nevertheless I want
you to spell it out, I will take a more constructive approach in our
discussion.

Hopefully this will lead to a dialogue that will, in turn, lead to
some progress in the field which, no doubt, is interesting.

ark

I/G to Ark:

Dear Ark,

We have received your message and the copy of your email addressed to
John Baez. We want to express you our gratitude for having been the
first to suggest that there might be some valuable ideas in our works.
We have read your comments about KMS : amongst all the mails, comments,
so call "analysis" etc, it appears that nobody has even an idea the
interest of the KMS condition. Your are the first to have raised the
fundamental property that KMS condition involves "analytic continuation
in time". The model on which we are working (including quantum
fluctuation of the signature and euclidean solution to initial
singularity problem) is based on this critically important KMS property.
With your help it might shed a totally new light on physics at the
Planck scale and provoke a dramatic adjustment in the setting of string
theory. Another fascinating item you have raised is this "symmetry
between positive and negative energy". This involvement of KMS condition
in such a symmetry is simply fascinating. Because -as you know- it
addresses one of the most intriguing problems of the standard big bang
theory : the matter/antimatter symmetry during the initial cosmological
phase at the Planck scale. The question (why matter gained against anti
matter? ) might find an answer in the framework of KMS theory.

But the existence of the matter/antimatter problem seems to be a very
important clue in favor to the idea of applying KMS condition at the
Planck scale.

(...)

Best

Igor / Grichka

Very happy New York Times has establish contact with you. We really
think that your opinion as one of the rare expert in the field is going
to modify the dominant opinion that is evidenced by the paper of
Chronicle of Higher Education. From a journalistic point of view, it is
much more "spectacular" to make the reader beleive that there was a
"coup" behind all this (and that two TV stars could fool all the whole
physics community) instead of a real work that might contain some
valuable ideas.

Strange...Grichka and I were discussing this afternoon about what
orinally happened to Tomita's work. Everybody said at that time that the
paper he wrote was pure non sense, worthless pages, etc. There was
only one person who looked a bit more carefully on the paper and noticed
that in spite of all the mistakes and errors, this work "contained
something". Therefore he decided to work on the paper in order to
extract the original idea out of it. This "rebuilder" was Takesaki and
the result of his action was the famous Tomita Takesaki paper that
originated one of the most powerful maths theories.

In a way (without having the pretention of being Tomita) we have the
feeling that to a certain extent his case presents some similarities
with ours.

Best regards,

On 7 Nov 2002 at 20:31, igor.bogdanov wrote:

> Strange...Grichka and I were discussing this afternoon about what
> orinally happened to Tomita's work. Everybody said at that time that
> the paper he wrote was pure non sense, worthless pages, etc.

Right. Takesaki took the pain of making sense of it.

> There
> was only one person who looked a bit more carefully on the paper and
> noticed that in spite of all the mistakes and errors, this work
> "contained something". Therefore he decided to work on the paper in
> order to extract the original idea out of it. This "rebuilder" was
> Takesaki and the result of his action was the famous Tomita Takesaki
> paper that originated one of the most powerful maths theories.

Indeed. It could have take 50 more years (or "never") to evolve the
theory by ordinary "constant pressure" effect.

> In a way (without having the pretention of being Tomita) we have the
> feeling that to a certain extent his case presents some similarities
> with ours.

The devil is in the details though :-)

ark

The "Constructive Discussion" will now proceed as bog-werk