The Bogdanoff Affair

John Baez

October 22, 2010


A Hoax?

We all laughed when the physicist Alan Sokal wrote a deliberately silly paper entitled Transgressing the boundaries: towards a transformative hermeneutics of quantum gravity, and managed to get it accepted by a journal of social and cultural studies, Social Text.

But in 2002, on the 22nd of October many of us began hearing rumors that two brothers managed to publish at least five meaningless papers in physics journals as a hoax - and even got Ph.D. degrees in physics from Bourgogne University on the basis of this work!

The rumor appears to have begun with an email from the physicist Max Niedermaier to the physicist Ted Newman, and it spread like wildfire. I received copies from many people, and soon there was a heated discussion of what this meant for the state of theoretical physics. Had the subject become so divorced from reality that not even the experts could recognize the difference between real work and a hoax?

On October 23rd I decided to post an article about this to sci.physics.research, a physics discussion group I help moderate. Entitled Physics bitten by reverse Alan Sokal hoax?, it brought widespread attention to the Bogdanoff affair. It also started a a fascinating discussion on sci.physics.research, to which Sokal and the Bogdanoffs themselves eventually contributed.

By October 24th, Dennis Overbye, a science journalist from the New York Times, began looking into this story. He asked the Bogdanoffs if their work was a hoax, and they indignantly denied it. Igor Bogdanoff soon began circulating emails to this effect. Again, these spread like wildfire, and most people interested in this case have already read one or another version.

Max Niedermaier then emailed the Bogdanoffs an apology, which he urged them to distribute.

Indeed, many aspects of the original rumor are known to be in error. According to Niedermaier, both Bogdanoffs conducted their thesis defense on the same day, in a rented hall with TV crews present. In fact, it seems clear that the Bogdanoffs got their Ph.D.'s at different times. Grichka got a Ph.D. in Mathematics from the Universite de Bourgogne on June 26 1999, passing at the lowest level. On the same day, Igor's thesis committee failed him. He later got a Ph.D. in Physics from the Universite de Bourgogne on July 8, 2002.

However, I assure you that the Bogdanoff's theses are gibberish to me - even though I work on topological quantum field theory, and know the meaning of almost all the buzzwords they use. Their journal articles make the problem even clearer. You can easily get ahold of these, because they are appended to the PDF files containing their theses. Some parts almost seem to make sense, but the more carefully I read them, the less sense they make... and eventually I either start laughing or get a headache.

For example, here's the beginning of Igor Bogdanoff's paper "Topological Origin of Inertia":

The phenomenon of inertia - or "pseudo-force" according to E. Mach [1] - has recently been presented by J. P. Vigier as one of the "unsolved mysteries of modern physics". Indeed our point of view is that this important question, which is well formulated in the context of Mach's principle, cannot be resolved or even understood in the framework of conventional field theory.

Here we suggest a novel approach, a direct outcome of the topological field theory proposed by Edward Witten in 1988 [3]. According to this approach, beyond the interpretation proposed by Mach, we consider inertia as a topological field, linked to the topological charge Q = 1 of the "singular zero size gravitational instanton" [4] which, according to [5], can be identified with the initial singularity of space-time in the standard model.

It goes on to discuss the supposed connection between N = 2 supergravity, Donaldson theory, KMS states and the Foucault pendulum experiment, which he claims "cannot be explained satisfactorily in either classical or relativistic mechanics". If you know some physics you'll find this statement odd. The Foucault pendulum behaves exactly the way classical mechanics predicts: it is a standard textbook exercise!

After several pages he concludes:

We draw from the above that whatever the orientation, the plane of oscillation of Foucault's pendulum is necessarily aligned with the initial singularity marking the origin of physical space S3, that of Euclidean space E4 (described by the family of instantons Ibeta of whatever radius beta), and, finally, that of Lorentzian space-time M4.
Zounds! He took that pendulum and rode it right off into hyperspace! I appreciate the fact that to someone not expert in physics, this stuff may seem no weirder than any other paper in a physics journal. He is indeed using actual physics jargon - but I assure you, it makes no sense. How in the world could the plane of oscillation of a pendulum be "aligned with the initial singularity", i.e. the big bang? The big bang did not occur anywhere in particular; it happened everywhere.

Indeed, nothing in any of the Bogdanoff's papers suggests that they really understand N = 2 supergravity, Donaldson theory, or KMS states. I'm reasonably familiar with all these topics, and as far I can tell, all they write about them is a mishmash of superficially plausible sentences containing the right buzzwords in approximately the right order. There is no logic or cohesion in what they write.

The Bogdanoffs posted an article on sci.physics.research arguing against my characterization of their work. You can read some of the subsequent dialog on my website, and more on sci.physics.research. George Johnson described this discussion quite well in the New York Times: he said that reading it is "like watching someone trying to "nail Jell-O to a wall". The discussion eventually fizzled out amid the Bogdanoff's attempts to justify their misuse of basic mathematical terminology.

Since their papers make no sense, what was their intention in writing them? By now it seems clear that they were not staging a Sokal-type hoax to show up defects in the refereeing and Ph.D.-granting process. They have lost too much face for this to be a plausible course of action! The main remaining options are 1) that they are engaged in some sort of trick, or 2) that they honestly believe in what they're doing.

Since they say they believe in what they're doing, this seems like a plausible explanation. However...

The Plot Thickens

On October 30th, I heard a strange story about the Bogdanoffs that suggested some complicated trickery was afoot. On November 5th, an article by Rich Monastersky appeared in The Chronicle of Higher Education, recounting this story:

Trinh X. Thuan, a professor of astronomy at the University of Virginia, sued the Bogdanoffs in France a decade ago, charging that, in God and Science, they had plagiarized his book The Secret Melody: And Man Created the Universe (published in English by Oxford University Press in 1995). In the end, he says, the judge found in his favor, and the brothers, along with their publisher, had to pay 80,000 francs to Mr. Thuan. The brothers say that they did not commit plagiarism and that Mr. Thuan had copied their own earlier work, so they never paid him any money.

The plagiarism lawsuit may explain why the twins were so eager to get doctorates, says Mr. Thuan. The back cover of their book claimed that they held doctorates when they did not, and they hurriedly tried to get degrees as the court case played out in the early 1990s, he says.

John D. Barrow, a professor of mathematical sciences at the University of Cambridge, says the brothers contacted him at that time with an odd request. "They were very anxious to obtain Ph.D.'s very quickly, and they tried to con me into becoming an examiner," he says. "There were two theses that they had submitted. They were laughable compendiums." As for the brothers, he says, "I regard them as mysterious people, not as hoaxers."

The Bogdanoffs say that the statement on the back cover of the book was the fault of a "clumsy" editor who wrote that they had degrees when they were actually in the process of earning them. They also deny trying to get doctorates quickly and say that they had contacted Mr. Barrow about long-term plans.

Such arguments between the Bogdanoffs and others do not surprise Jean Staune, general secretary of the Interdisciplinary University of Paris, who helped Mr. Thuan with the plagiarism case against the brothers. "They are like water," he says. "You can never catch them."

(In the original version of this article, John Barrow had said " I regard them as sinister people". But then the Bogdanoffs threatened to sue, so this was changed to "I regard them as mysterious people".)

The Bogdanoffs come across rather badly in this article. However, their thesis advisor, Daniel Sternheimer, tells the story quite differently in his response to Monastersky's article:

Dear Monastersky,

I have been forwarded the text of your November 5 Chronicle article, entitled "French TV Stars Rock the World of Theoretical Physics". I shall make only a few comments on facts.

1. You write:

"According to Mr. Sternheimer, Grichka Bogdanoff applied for his Ph.D.in physics in 1999 but was granted one in mathematics instead"
There must be a misunderstanding. Our group (UMR 5029) is in mathematical physics, and its members are affiliated with the Department of Mathematics of Universite de Bourgogne or with the Section of Mathematics in CNRS (or visitors and students). I was sole (administrative) supervisor for Grichka in Dijon. In 1999, given his Thesis, though we have a common "Ecole Doctorale" at our UFR ("faculty") Sciences et Techniques, it was clear that Grichka's thesis was in mathematics and that appears on the cover of the final Ms. In both cases there was only one other professor from Dijon to complete the Jury: for Grichka, it was a mathematics professor; for Igor, it was a theoretical physics professor: that is not coincidental.

2. You write:

"Trinh X. Thuan, a professor of astronomy at the University of Virginia, sued the Bogdanoffs in France a decade ago, charging that, in God and Science, they had plagiarized his book The Secret Melody:And Man Created the Universe (published in English by Oxford UniversityPress in 1995). In the end, he says, the judge found in his favor, and the brothers, along with their publisher, had to pay 80,000 francs to Mr. Thuan. The brothers say that they did not commit plagiarism and that Mr. Thuan had copied earlier work of their own, so they never paid him any money."
"TXT" is certainly a well established astrophysicist and he is also a quite successful writer of essays, even if none of his essays had the success of "Dieu et la Science". But his memory is very partial.

It is true that a first court decision (in "refereť" in 1991, confirmed in 1992) gave Fayard and him a modest provision of 50000 FFR (50kF) from Grasset and 15kF from the twins. But in January 1994, four members of the French Academie des Sciences (Yves Coppens, Jean Dorst, Andre Lichnerowicz, Etienne Wolf) issued an affidavit, confirmed by Jacques Friedel, then President of the Academy, giving among others a non exhaustive list of 9 quotations, 5 of them appearing in "La melodie secrete", which come from texts by Guitton or the twins prior to the latter (from 1963, 1986 and early 1987).

Prior to that, the magazine Paris Match had given, in its September 12, 1991, and confirmed in affidavits from October 1993 and January 1994, examples proving that TXT had reproduced parts of a "Grand Document" of 50 typewritten pages by Guitton and the twins (December 1986) on the place of God in modern science, and some of another (January 1987) based on the transcript of an interview of Carl Sagan by the twins.

After that a first court decision in March 1994 awarded only 60kF+20KFto TXT for some technicalities. The whole matter of respective accusations of plagiarism went on, but was eventually stopped short by Hachette and an out-of-court agreement was reached, according to which the provisions were returned, TXT renounced to the 80kF, each side paid for the court and lawyer expenses it had made, and the following press release was issued by Hachette (the translation is mine, and I cut some civilities and full names):

"Regretting the echos in the media of the dispute between them, Fayard and their author TXT, Grasset and their authors Guitton and Messrs Bogdanoff, have decided to put a final stop to it, leaving to the whole scientific community the expressions and metaphors to which both of them referred, preserving in this way the respective integrity of the authors."
I have copies of original documents proving the above facts.

The discrepancies between what TXT told you and the last agreement, which he signed along with all parties involved on May 13, 1995, are (mildly speaking) strange. They are far more serious than many menial errors or misrepresentations that the twins let pass (like some about their age, which I have seen in a magazine quoted as 38 when theywere 50; that was 2 years ago; like many women, TV stars feel often flattered to be thought younger than they are).

Another fact is that the "Secret Melody" is now being re-edited and should be out in December (I found that with "Google"). Still another is that "Dieu et la Science" was never published in English (though it has been translated in many languages), and that lawyers' fees in the US can exceed by far any profit from translations. Any connection between these facts is surely coincidental.

From a distance such as mine, it's quite hard to tell what's really going on. Perhaps some long-standing enmities are the ultimate cause of some aspects of this fuss - in which case it may be almost impossible to unravel all the details.

The Media Uproar

When the Bogdanoff affair hit in the popular press, it caused quite a stir. Has theoretical physics has become so abstruse that nobody is able to tell a hoax from serious work? - this theme proved almost irresistable to the journalists.

On November 1st, Andrew Orlowski wrote an amusing but essentially frivolous account of the whole affair in the British paper The Register, entitled Physics hoaxers discover a quantum bogosity.

On November 7th, Christoph Droesser and Ulrich Schnabel wrote a story about it in the German paper Die Zeit, entitled Fairy tales of the brothers Bogdanov. This has some interesting quotes in it, which I will come back to later.

Also on November 7th, a story about the Bogdanoff affair appeared in Nature, written by Declan Butler. This really plays up the supposed uncertainty on the part of the physics community as to whether the Bogdanoff's work makes sense or not. For example, Butler writes:

So are the papers good science or not? Enquiries by Nature show that few theoretical physicists, including some who reviewed the brothers' Ph.D. theses, are completely certain. Jac Verbaarschot, of Stony Brook University in New York, reviewed Igor's Ph.D. He says it contained original ideas, but claims that it was awarded in part because of Igor's contributions to the public understanding of science. Others have come to harsher conclusions. "They were at best wrong, and most likely just throwing around words with no calculations or proofs to back them up," says Lee Smolin of the Perimeter Institute for Theoretical Physics in Waterloo, Canada, who has studied some of the papers.

But Roman Jackiw, a physicist at MIT who reviewed Igor's thesis, insists that it is of the requisite quality. Robert Coquereaux, from the International Centre for Mathematical Meetings in Marseille, has said that the brothers' work is certainly no better or worse than that of some established theoretical physicists.

The brothers, who are currently presenting a short television programme in France, insist that their work is genuine. They say that many critics haven't actually read their entire theses, which are available only in French, and that none of the criticisms made discredits their work. They also point to referees' reports on their theses. Verbaarschot, for example, declared that Igor's Ph.D. "ranks as one of the best I have seen in recent years".

With no clear consensus emerging, the credibility of the peer-review system and journals in string theory and related areas is taking a battering. Peter Woit, a mathematician at Columbia University in New York, says that the incident illustrates the speculative nature of much theoretical physics. "The Bogdanoffs' work is significantly more incoherent than just about anything else being published," he says. "But the increasingly low standard of coherence in the whole field is what allowed them to think they were doing something sensible and to get it published."

On November 9th, Charles Arthur wrote a story for the Independent entitled Did twins doctor up doctorates with physics buzzwords? There's nothing much new in this one.

Also on November 9th, Dennis Overbye came out with a story in the New York Times, entitled Are they a) geniuses or b) jokers? Like the story in Nature, this plays up the idea that physicists can't tell good work from bad. It contains quotes from various physicists criticizing the Bogdanoff's work, but also quotes defending it - especially from Daniel Sternheimer and Roman Jackiw. Sternheimer, you will recall, was the thesis advisor for both the Bogdanoff brothers. Jackiw, a professor of physics at MIT, was one of two `rapporteurs' who approved Igor Bogdanoff's thesis. Overbye writes:

Igor's thesis had many things Dr. Jackiw didn't understand, but he found it intriguing. "All these were ideas that could possibly make sense," he said. "It showed some originality and some familiarity with the jargon. That's all I ask."

Igor got his degree in theoretical physics from the University of Bourgogne in July, also with the lowest possible grade, one that is seldom given, Dr. Sternheimer said.

"These guys worked for 10 years without pay," he said. "They have the right to have their work recognized with a diploma, which is nothing much these days".

On November 17th, George Johnson wrote an article about the Bogdanoff affair in the New York Times, concluding that:

As the reverberations from the affair begin to die down, physicists seem to have accepted that the papers are probably just the result of fuzzy thinking, bad writing and journal referees more comfortable with correcting typos than challenging thoughts.

Dr. Sokal seemed almost disappointed. "If someone wanted to test a physics journal with an intentional hoax, I'd say, `more power to them'," he said. "What's sauce for the goose is sauce for the gander."

The Journals Respond

In the above quote, Johnson put his finger on an important issue: why did at least five journals accept the Bogdanoff's papers? I have obtained the referees' reports on three of the Bogdanoff's papers, which confirms that indeed, some referees were more interested in correcting minor typos than checking the logic of the papers. I also have a damningly negative report by another referee who is now willing to come forth publicly: Eli Hawkins. However, not enough people wrote reports like this to prevent the Bogdanoffs from publishing.

What did the journals say about it all? The Chronicle writes:

Mr. Wilczek is editor in chief of Annals of Physics, which published one of the Bogdanoff brothers' papers in February. But he and all of the current members of the journal's editorial board had recently joined and did not handle papers in that issue. He says that standards at the journal had slipped in recent years because of the illness and death of a previous editor in chief.

Although he will not comment on the Bogdanoff paper, Mr. Wilczek says he intends to raise the journal's standards. As part of that drive, members of the editorial board now do most of the reviewing. "I'm trying to get much tighter control, just because of things like this," he says, referring to the Bogdanoff case.

Hermann Nicolai, editor of Classical and Quantum Gravity, told Die Zeit that if the Bogdanoffs' paper had reached his desk, he would have immediately sent it back: "The article is a potpourri of the buzzwords of modern physics, that is completely incoherent."

Sometime around November 1st, the editorial board of Classical and Quantum Gravity issued the following statement, which I obtained through Greg Kuperberg:

Classical and Quantum Gravity and the paper "Topological theory of the
initial singularity of spacetime" by G Bogdanoff and I Bogdanoff, Class.
Quant. Grav. 18 4341-4372 (2001)

A number of our readers have contacted us regarding the above paper
and in response we have decided to issue the following statement.

Classical and Quantum Gravity endeavours to publish original research
of the highest calibre on gravitational physics. It is not possible for the
Editorial Board to consider every article submitted and so, in common
with many journals, we consult among a worldwide pool of over 1000
referees asking two independent experts to review each paper. Regrettably,
despite the best efforts, the refereeing process cannot be 100% effective.
Thus the paper "Topological theory of the initial singularity of spacetime"
by G Bogdanoff and I Bogdanoff, Classical and Quantum Gravity 18
4341-4372 (2001) made it through the review process even though, in
retrospect, it does not meet the standards expected of articles in this
journal.

The journal's Editorial Board became aware of this situation already in
April 2002. The paper was discussed extensively at the annual Editorial
Board meeting in September 2002, and there was general agreement that
it should not have been published. Since then several steps have been
taken to further improve the peer review process in order to improve the
quality assessment on articles submitted to the journal and reduce the
likelihood that this could happen again. However, there are at this time
no plans to withdraw the article. Rather, the journal publishes refereed
Comments and Replies by readers and authors as a means to comment
on and correct mistakes in published material.

We are also grateful to our readers, contributors and reviewers for their
vigilance and assistance both before and after publication.

Dr Andrew Wray
Senior Publisher
Classical and Quantum Gravity
Institute of Physics Publishing

Professor Hermann Nicolai
Honorary Editor
Classical and Quantum Gravity
Albert Einstein Institute

What Next?

The Bogdanoff affair seemed to run out of juice at the end of 2002. The scandal it caused in French academia was dealt with by a committee of professors who wrote a long report... which was then quietly filed away. (I know this from talking to one of the people who wrote it.) The media lost interest, and that seemed to be the end.

But more recently, it has reemerged!

In 2003 I received an email from a mysterious "Professor Yang" arguing that the Bogdanoff's work made sense. This fellow claimed to be from the "International Institute of Mathematical Physics" at Hong Kong University in Kowloon. I had never heard of him before, so I disregarded his email.

In the summer of 2003 I got an query from Dennis Overbye of the New York Times, saying that he'd gotten a similar email from Professor Yang, and asking me what was up. By sheer chance, I was spending this summer in Kowloon, Hong Kong! So, I was able to contact Hong Kong University and check that there was no International Institute of Mathematical Physics at this university, and no "Professor Yang" in their physics department!

Later, on December 29, 2003, a string theorist named Jacques Distler received a similar email from "Professor Yang", and was able to trace it back to France! He replied to the mysterious Yang; Yang emailed back, again from France... and the email was studded with grammatical errors typical of a native French speaker. At this point, it became fairly clear that Yang was none other than the Bogdanoffs in disguise! You can read all the details on Distler's web log.

The mathematician Peter Woit also got an email from the mysterious Yang on December 29, 2003. He later discussed it on his web log, pointing out the rather obvious evidence that it came from France. He got another email from one "Roland Schwartz", defending the Bogdanoffs, which he traced back to the same source.

Even better, on June 27, 2004 Woit noticed that the Bogdanoffs had set up a website purporting to belong to a Mathematical Center of Riemannian Cosmology, with a Latvian URL! This presumably nonexistent "center" claims to be devoted to research on the Bogdanoff work. You can read more about it on Woit's web log.

Why are the Bogdanoffs struggling to redeem their reputation by such underhanded yet comically inept methods? Are they just jokers?

No, not just jokers. They are still doing some TV shows in France, and they have come out with a book, Avant le Big Bang - or "Before the Big Bang" in French. It's full of silly errors, some which have been listed by Y. B. Messager. But, it seems to be selling quite well!

So, the brothers Bogdanoff may be managing to make money from their minor reputation as experts in France. That would explain a lot.

Still More

In February of 2005, Jean-Pierre Messager sent me the following news:
I noticed you updated your "Bogdanoff page" here is some news about this affair in France: Regards, JP. Messager
The cosmologist is Alain Riazuelo; he wrote a lot of comments about the Bogdanoff's stuff, most of which can be seen on a French mathematics forum hosted by the University of Strasbourg. The mathematicians include Damien Calaque, and their comments can be found on another forum.

In October 2010, the CNRS report on the Bogdanov brothers' theses — the one that had been "quietly filed away" — came to light. The weekly called Marianne found it somehow:

For more details see Peter Woit's Bogdanov Update dated October 18th, 2010.

The Wikipedia Article

There is an excellent Wikipedia article on the Bogdanoff Affair.

For quite some time, and as recently as June 2006, the Bogdanoff brothers and a large crowd of sock puppets have been attempting to edit this article to make it less embarrassing to them. For details, see the talk page. Nobody seems to be fooled....

Decide For Yourself

If you wish to judge the Bogdanoff's work for yourself, there is nothing to do but master the necessary physics and read what they have written. Their theses are available in PDF format online - at least for now:

The Bogdanoff's theses are in French, but there are English versions of the abstracts. Here's the abstract of Igor Bogdanoff's thesis:

We propose in this research a new solution regarding the existence and the content of the initial spacetime singularity. In the context of topological field theory we consider that the initial singularity of space-time corresponds to a zero size singular gravitational instanton characterized by a Riemannian metric configuration (++++) in dimension D = 4. Connected with some unexpected topological data corresponding to the zero scale of space-time, the initial singularity is thus not considered in terms of divergences of physical fields but can be resolved in the frame of topological field theory. We get this result from the physical observation that the pre-spacetime is in a thermal equilibrium at the Planck scale. Therefore it should be subject to the KMS condition. We consequently consider that this KMS state might correspond to a unification between "physical state" (Planck scale) and "topological state" (zero scale). Then it is suggested that the "zero scale singularity" can be understood in terms of topological invariants, in particular the first Donaldson invariant. Therefore, we here introduce a new topological index, connected with 0 scale, of the form Zbeta = 0 = Tr (-1)s, which we call the "singularity invariant". Interestingly, this invariant corresponds also to the invariant topological current yielded by the hyperfinite II* von Neumann algebra describing the zero scale of space-time. In such a context we conjecture that the problem of inertial interaction might be explained in terms of topological amplitude connected with the singular zero size gravitational instanton corresponding to the initial singularity of spacetime.

Here's the abstract of Grichka Bogdanoff's thesis:

We propose hereafter that the signature of the Space-Time metric (+++-) is not anymore frozen at the Planck scale and presents quantum fluctuations (++++/-) until 0 scale where it becomes Euclidean (++++). (i) At the albraic level we suggest an oscillation path (3,1) (4,0) excluding (2,2). We built the quotient topological space describing the superposition of the Lorentzian and the Riemanian metrics. In terms of quantum groups we evidence a relation between q-deformation and deformation of the signature. We have obtained a new algebraic construction (a new cocycle bicrossproducts by twisting) which allowed us to unify the Lorentzian and the Euclidean signatures within a unique quantum group structure. Moreover the q-deformation of space-time shows that the natural structures of q-Minkowski and q-Riemanian spaces are linked by semiduality. (ii) Regarding the physical motivations we suggest that at the Planck Scale the Space-Time is in KMS state. Within the limits of the KMS holomorph strip, the beta timelike parameter is complex. We propose an extension of relativistic gravity which begins at the Planck Scale with the Lagrangian R + R2 + RR*. Then, the infrared limit of the theory is given at the Planck Scale by the Einstein term in R and corresponds to the Lorentzian metric while the ultraviolet limit is given at beta=0 scale by the topological term RR* and corresponds to the Euclidean metric ( topological sector). We propose a duality between instantons and monopoles in 4 dimensions giving a representation of the superposition of the metrics. (iii) On the cosmological plan we suggest to describe the Initial Singularity of Space Time by a topological invariant I(S) = Tr(-1)S which is analog to the first Donaldson invariant. The initial singularity must be considered as a singular 0-size gravitational instanton. The physical observables are therefore replaced by cycles of homology in the moduli space of gravitational Instantons. We propose a conjecture regarding the existence of a topological amplitude associated to a "topological expansion phase" which preceeds the classical cosmological expansion. This topological phase is also able to be described by the flow of weights of the II* hyperfinite factor type corresponding to the beta=0 initial singularity.

The papers by Grichka and Igor Bogdanoff include:

I have looked at all these papers. It is interesting to note that the Annals of Physics paper is almost identical to the Nuovo Cimento paper, and the Chinese Journal of Physics paper is also very similar to these two.


© 2010 John Baez
baez@math.removethis.ucr.andthis.edu

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