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This week 208: analysis

I am just reading "This Week's Finds in Mathematical Physics" number 208 by John Baez. Powerful stuff.

John Baez describes a conference at the Perimeter Institute. What was the topic? Well, it was obviously loop quantum gravity, but the organizers chose a pretty self-confident title that includes the words "Quantum Gravity in the Americas". Wow.

These conferences seem to be a spiritual continuation of the conferences decades ago, such as the 1962 general relativity conference in Warsaw. In a letter to his wife, Feynman wrote:

  • I am not getting anything out of the meeting. I am learning nothing. Because there are no experiments, this field is not an active one, so few of the best men are doing work in it. The result is that there are hosts of dopes here (126) and it is not good for my blood pressure. Remind me not to come to any more gravity conferences!

Let's mention that Feynman himself derived the Feynman rules for general relativity in 1963. He also showed that the tree diagrams agree with the classical theory. Because of these and other contributions, it is clear that his opinion about quantum gravity mattered.




At the beginning of his essay, Baez describes the institute as a physicist's heaven, and explains that he gave the first talk because Abhay Ashtekar got lost in the new building. In his talk, Baez enumerated all recent papers about loop quantum gravity. I've heard about most of these papers, and there is probably nothing interesting to talk about. However it's already the "abstract" of John Baez's program that seems highly problematic - sort of shocking. One might say that it summarizes what Baez considers to be the main task for quantum gravity:

  • The problem of dynamics in quantum gravity is still a big challenge. We don't know how to make spacetime into a truly dynamical entity with local degrees of freedom while taking quantum theory into account. Neither string theory, nor loop quantum gravity, nor the spin foam and causal dynamical triangulation approaches have yet found a background-free quantum theory with local degrees of freedom propagating causally. We sketch some avenues for making progress in this direction.

Wow. To understand dynamics in quantum gravity, according to John Baez, means to describe spacetime with local degrees of freedom that propagate causally. This is, according to Baez, what people interested in quantum gravity should work on.

It's not presented as a modest speculative proposal about a possible new description of quantum gravity, but rather as a universal key to judge the success of any enterprise in theoretical physics.

Well, there is a growing body of theoretical evidence that indicates that while the principles of quantum mechanics will survive in our future theories of quantum gravity without any modifications, the geometric concepts, including causality and locality, will not. As our understanding of quantum gravity deepens, it seems increasingly likely that geometry - as well as the related concept of locality - is a derived, approximate concept that follows from a much more rich theory, one that does not respect the naive ideas about geometry.

Non-locality or a subtle violation of causality is what seems to be an ingredient of the most likely resolution of the black hole information puzzle. String theory shows us a lot of new very specific phenomena that modify our ideas about geometry at very short distances, and many interrelations between the concepts that we usually associate with "geometry" and those that we usually associate with "matter". We've learned quite a lot about geometric transitions, dualities, and so forth.

Cargo cult science

OK, most of John Baez's assumptions what a theory of quantum gravity should look like seem relatively unlikely, and sort of obsolete. But even if we accept the idea that it is fine for a physicist involved with quantum gravity to be ignorant about string theory, including its very basic aspects, Baez's approach won't be really scientific because it is similar to the cargo cult.

There exist tribes at the Vanuatu archipelago, described by Feynman, that have chosen magicians with wooden "earphones", and they expect the airplanes to land, much like they were landing during the Second World War, and bring them a lot of nice stuff. Everything looks perfect but something must be missing because the airplanes do not land. It may be rather difficult to explain these tribes what's wrong with their science: they would prefer if you told them how should they modify the shape of their wooden earphones.

Baez's approach to quantum gravity is analogous. The starting point is easy because he already knows the answer to the main questions of quantum gravity: there should be exactly local and causal degrees of freedom that propagate through a "background free" spacetime. (These are the earphones.) Incidentally, the phrase "background free" does not really reflect anything reasonable, beautiful, or justified. According to the LQG ideology, string theory as such is not background free. If you discuss this topic with the LQG proponents, you won't learn what "background freedom" exactly is, but you will definitely end up with the same feeling as me - namely that it is some sort of Mach's principle, i.e. something that is known to be false.

OK, so we have the earphones. The only task that remains is to find the right variables that express the data about the metric, and the right rules how to deal with these variables. (This is the shape of the wooden earphones.) Today, this idea may look even more naive than Einstein's rather naive ideas about the form of the unified theory.

Baez does not find it important to verify experimentally - or at least by a more detailed, quantitative theoretical calculations or arguments - whether his assumptions are correct and sufficient, or at least predictive. The important answers are known a priori, and the task for a physicist is to accept these assumptions as dogmas and try to find evidence. Sorry, but this attitude is analogous to creationism, and it is pseudoscientific, especially if someone continues with this approach after more than 40 years of failed attempts to obtain anything from this approach that goes beyond classical GR, and could be compared with experiments.

Particles as wormholes

When I was 15, I thought I had a perfect theory of elementary particles. First of all, at that time I believed that there were only three truly elementary particles - the electron, the neutrino, and the proton, and perhaps their antiparticles. All of them were modeled as topological defects in spacetime.

Imagine that you remove two balls from the 3-dimensional space, and identify the boundaries of these two balls. Well, then you obtain the electron. If you remove two "solid doughnuts" and identify the 2-toroidal boundaries, you will end up with a neutrino. If you remove two "solid genus 2 Riemann surfaces" and identify them, you will obtain a proton. Moreover, the genus 2 Riemann surface may be chosen to have a S3 symmetry, which - I thought - should explain the fact that it seems that there are three quarks inside the proton.

I did not know what the higher genus wormholes meant. Probably new particles? ;-)

Well, it would probably be quite difficult for me to advocate this theory today. The immediate reason why I discarded the theory at that time was my inability to describe the annihilation as a smooth process - today, I would have other reasons to be suspicious. Nevertheless, it seems to me that John Baez is trying to do more or less exactly this thing. What's wrong with these theories? Naively, they can be attractive, but after 5 minutes, if you try to reconcile them with any "details" we know, you will see that no detail about them can work - certainly not quantitatively. Moreover, once we have quantum mechanics, there is a plenty of new ways how new physics emerges (new massless states arising from geometries that look singular classically, which is a common phenomenon in string theory), and the idea that all known elementary particles must be associated with a particular smooth geometry is much less attractive in a quantum mechanical world than it would be in a classical world.

The difference between the string theoretical approach and this loop quantum gravity approach can be summarized by another observation: string theory is focusing on the mathematical structures that can lead (and do lead) to predictable and sufficiently unique physically (semi-)realistic outcomes that morally resemble the other things we know must be there in a physical theory, to say the least. Loop quantum gravity and similar approaches, on the other hand, emphasize naive classical pictures of reality (such as LEGO, spin foams, particles as wormholes, cellular automata), and it tries to "prove" that they are relevant for the Universe.

You might think that a string is just another object, much like a wormhole, a cell in a cellular automaton, or a piece of spin network. But we don't study string theory because we like the shape of the string or because it looks simple to us. We study it because the two-dimensional theory describing stringy worldsheets is conformal, which allows us to eliminate all local metric degrees of freedom on the worldsheet (it's therefore a renormalizable 2D theory of gravity), and reduce the path integral over the Riemann surfaces (histories of strings) into a finite-dimensional convergent integral whose results moreover give us a unitary S-matrix that reproduces all qualitative phenomena that we know from GR and gauge theories.

If the strings could not do it, we would definitely avoid their investigation.

In other words, string theorists start with the question - only a scientific one that can a priori have several different answers - and then try to find calculations or solid arguments, and these calculations eventually support one particular answer - which often forces them to enrich their mathematical toolkit, the ideas about "what is natural to expect", and the warehouse of nice ideas. The result, happily, happens to be a unique theory that unifies all these ideas, namely string/M-theory, and we are learning new stuff from it.

Loop quantum gravity guys start with a constant collection of ideas and an answer - that has been revealed to them by a divine power - and then try to show that the answer must be correct even though most of these answers are quite obviously incorrect. More generally, the "discrete people" (see the sci.physics.discrete newsgroup or Wolfram's book) start with the dogma - the most important insight about the Universe is that it is discrete. The only task that remains, according to them, is to fill in the details of their ridiculous model of the Universe.

Obviously, this approach is very unlikely to work in science. It's not surprising that this approach prevents its advocates from seeing the truly interesting ideas - those that only emerge if one is ready to admit that he or she does not know everything from the very beginning, and he or she is ready to learn new things, either from the experiments, or from unbiased, mathematically deep theoretical research of some well-defined and interesting theories, or from others. Ideas such as renormalization group, confinement, holography, geometric transitions, enhanced gauge symmetries, dualities, and many others.

Contact with observable physics

Let's return to Baez's week. Baez continues with a discussion of some random papers - it seems that he must have a policy that he would never choose a paper that has led to some progress, e.g. a paper with at least 10 citations. The main reason why I find these attempts hopeless is undoubtedly the same reason as why Feynman did not like this stuff. They just don't care about making a contact with observable physics.

If we study gravity, it seems likely that classical general relativity is sufficient for our understanding of all phenomena at pretty long distances (it may, conceivably, break down at cosmological distances or tiny accelerations, but let's not discuss this possibility here). If we want to reveal the quantum influences on gravity, we are more or less inevitably thinking about physics at short distances, which are usually tested by high-energy particle physics (e.g. accelerators). The language of effective field theory, cross sections, amplitudes, and so forth is unavoidable.

Particle physicists and string theorists study many different ideas - some of them are combinatorial and "discrete", most of them are not - but in all these cases, they must be equivalent or continuously connected to the "ordinary" types of physical theories - those that can be verified in actual experiments. These connections are necessary for any research to be called "physics". If someone is inspired by a symphony, by a tiger's skin, a principle of the philosopher Ernst Mach, LEGO, or anything else, and she or he tries to claim that these ideas underlie all of physics, such a statement is not yet physics and most likely, it never will.

Progress since 1960s

John Baez describes various random speculations about the ideas that might be relevant for quantum gravity sometime in the future - such as "gravity as perturbation of a topological field theory" or "treating 4D gravity much like 3D quantum gravity, even though the latter has no local excitations". He also tries to explain "dark matter using quantum gravity", before he makes any attempt to explain the known matter (such as quarks, leptons, and gluons). It seems pretty clear that most of these ideas could have been proposed 50 years ago, and they are not really affected by any solid insights in physics made in the last 50 years (and there have been very many!). In the most optimistic cases, the authors may speculate that Newton's laws from the 17th century might possibly emerge from some crazy structure that needed to be fine-tuned and modified by several unjustified procedures.

It looks obvious that this type of research is stuck in a closed loop and the rate of progress equals zero. I am amazed by the nerves that the people must have to study physics in this way. The only reason why physics research is meaningful even today is that we are doing things that go beyond the things that were known to the physicists 50 or 250 years ago - at least sometimes we are doing such things. The progress in the 1920s may still have been more profound than in the 1990s, but the recent progress is not negligible.

But some dogmatic ideas about the discrete structure of spacetime - whether it's A New Kind of Science or Loop Quantum Gravity just seem like randomly imported ideas from the era of aether, or from the contemporaries of Democritus. They clearly constitute a very negative progress if we compare it to the Standard Model or General Relativity: they don't even seem to have a chance to describe the known phenomena from the SM and GR - and their research is even more disconnected from the cutting-edge experimental physics that will give us new information which paths beyond the SM and GR - e.g. which scenarios in string theory - are realistic.

The fact that a nonzero fraction of the theoretical physicists is working on something that is so obviously less interesting than science studied by the generation of physicists 35 years ago makes me a bit depressed, but I can imagine that physics always looked like that.

At least, the rest of us is constrained by all the known experimental facts about GR and the Standard Model, and those of us who try to look for new phenomena, new dual descriptions, and new language for string theory are also constrained by the large body of knowledge we have about string theory - a theory that is connected to all physics of the previous theories. But I just don't understand how someone can try to construct a theory of physics from the scratch, without looking at the known detailed facts about physics (that certainly go much beyond the clichés about background freedom), and how should we distinguish this research from the research of other people who try to start from the "scratch", those that are usually called crackpots.

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reader Anonymous said...
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reader Lumo said...
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reader Plato said...

This seems to be the way of it , it to refute all positions taken.

I thought I would leave a link here like I did at Peter's Blog for the Landscape of the Week, picture.:)

Your recipe for holographical perspective of 2d work in branes, just seem natural to me as shadows on the wall. Hooft makes use of this picture too?:)

Lubos...We study it because the two-dimensional theory describing stringy worldsheets is conformal, which allows us to eliminate all local metric degrees of freedom on the worldsheet (it's therefore a renormalizable 2D theory of gravity), and reduce the path integral over the Riemann surfaces (histories of strings) into a finite-dimensional convergent integral whose results moreover give us a unitary S-matrix that reproduces all qualitative phenomena that we know from GR and gauge theories.We just had to recogize the dimensional sigificance of moving to one dimension lower? Looking at the "sunlight" we remove the constraints of seeing with our new perspectives?


reader Lumo said...

Hi Plato! Landscape of the week is pretty funny ;-).

Well, yes, I think that 't Hooft uses shadows on the wall a lot (including shadows of his Powerpoint presentations). :-) But you know, a shadow is not yet a hologram.

By the way, moving one dimension lower may look arbitrary, and indeed, in some cases we want to argue that one can move many dimensions lower.

But moving one dimension lower is special because the value of the entropy (amount of disorder, or maximal information in bits carried by) of the black hole scales like its surface - which always has one less dimension than the bulk. It just seems to be a fact of quantum gravity, and this fact makes it natural to define a theory on the surface, and the AdS/CFT correspondence is a very well working example of it.

Have you seen an actual hologram? It's a sticker that is completely two-dimensional, but when you look at it, it looks entirely three-dimensional. It's because of some interference going on in it, and we believe that something analogous takes place in quantum gravity. It's not just a shadow - it's a very fine picture with tiny strips whose wavelengths etc. encode the 3+1 dimensional information.


reader Anonymous said...

I agree with almost all of your comments. But I just don't see why they are not applicable to string theory. Can we really say that stringy picture has more connection with experiments than LEGO picture? Can string theory will give us something we already don't know, some day? Unless it calculates electron charge/mass ratio or number of families etc. it doesn't have much weight. Kaluza-Klein theory enables us to write GR and E&M in the same language, but even if there was no nuclear forces it would not be "the" theory of everything unless it gives something observable more than GR+E&M.
I feel like you are applying different standards to string theory and everything else. Can you elaborate more on that?


reader Arun said...

Lubos,

As far as I know, John Baez is a mathematician, not a physicist. So he doesn't have to make any contact with reality. Perhaps your comments about the other participants and the relevance to physics may hold good. I'll take your word for it that none of the papers Baez mentioned is worth reading :)

-Arun


reader Lumo said...

Hey Arun! You must look at the papers yourself, I may have missed one which will become very important! :-)

John Baez calls himself a mathematical physicist, I thought, does not he? ;-)

To the previous contributor: Yes, string theory has the power to calculate the matter content and masses of particles in a background once you identify this background.

We don't know the right background yet, and therefore we can't calculate the specific masses seen in this Universe.

But even if you forget about the ability to predict the masses, there is still a pretty big difference between string theory that CAN reproduce the Standard Model and gravity on one side, and LEGO that *cannot* reproduce them on the other side, don't you think??

I assure you that string theory is the only known theory that is not a theory of the same type as the previous theories - a quantum field theory like the Standard Model - but that can reduce to them.

I think that I am applying absolutely the same criteria to all theories. If you don't think so, could you please be more specific?

If you don't consider the ability of a theory to reproduce at least the previous successful theories to be an important feature, it may be however hard for two of us to achieve a consensus.


reader Matti Pitkanen said...

Lubos produced a magnificent piece of rhetoric. There is
however the danger that the reader forgets that the target was LQG rather than M-theory(;-). Because I have an somewhat ambivalent relationship with M-theory and well aware about the traumatic relationship of M-theory and experimental reality, I have some difficulties remembering who was the enemy.


I read This Week's Finds and found interesting links to the quantization of Teichmueller spaces, which I regard as a fascinating topic with a possible relevance for the p-adic variants of string models. My own interest stems from the fact that the requirement that Topological Geometrodynamics allows an algebraic continuation to all number fields, has turned a very
powerful constraint allowing to deduce highly non-trivial predictions. p-Adic mass calculations, which I performed for the first time for more than decade ago, provide an excellent example of this. Quite recently I have been working again with the closely related quantization of conformal moduli/Teichmueller parameters following from the algebraic universality of physics.


1. Partons as 2-surfaces

In TGD framework partons correspond to 2-D surfaces X^2 whose "orbits" are light like 3-surfaces X^3_l and allow by their metric 2-dimensionality generalized conformal invariance. X^3_l in turn act as causal determinants for 4-D space-time surfaces in M^4xCP_2 in the spirit of quantum holography and general coordinate invariance.


2. Elementary particle vacuum functionals


Elementary particles are characterized by elementary particle vacuum functionals in the space of conformal moduli of X^2 forming a finite-dimensional space parametrized by Teichmueller parameters. See

http://www.physics.helsinki.fi/~matpitka/tgd.html#elvafu

Elementary particle vacuum functionals are expressible using products of theta functions of even characteristic in the moduli space of 2-surfaces: kind of a mini-super space for 2-dimensional worlds. Simple physical requirements besides modular SL(2g,Z) invariance fix the ground state elementary particle vacuum functionals uniquely and an explanation for why
only genera g=0,1,2 (e,mu,tau) correspond to light elementary particles emerges in terms of hyper-ellipticity. Maxima of Kahler function correspond to hyper-elliptic partonic 2-surfaces allowing Z_2 conformal symmetry and elementary particle vacuum functionals vanish for them for genera g>=2 since some the theta functions vanish identically for hyper-elliptic 2-surfaces.

3. Modular contribution to particle masses

p-Adic mass calculations predict a contribution in particle mass squared expressible formally as a thermal expectation value of a "thermal conformal weight" over conformal moduli characterizing partonic two-surface. See previous link and

http://www.physics.helsinki.fi/~matpitka/padtgd.html#mass1
.

This partition function is the square of elementary particle vacuum functional, which itself is a product of "partition functions". This alone allows to deduce the general dependence of the contribution on genus g.


4. Number-theoretical quantization of moduli


The great philosophical idea of p-adic mass calculations is that p-adic thermodynamics for Virasoro generator L_0 (and entire quantum TGD) is algebraically continuable from real context to p-adic number fields. Physics is algebraically universal. This quantizes the temperature parameter associated with Virasoro generator L_0 and mass scales are quantized in
terms of prime p. Primes p=about 2^k, k prime have turned out to be of a special physical relevance and given elementary particle is characterized by the prime k. For instance, electron corresponds to Mersenne prime 2^127-1 and thus for k=127.

Similar algebraic continuation must be possible for the modular contribution to mass. Algebraic continuation however poses two problems. p-Adic variants of theta functions must exist p-adically and the integral over moduli must be well defined in p-adic sense.

Theta functions must exist p-adically. That is, the exponentials appearing in the infinite sums defining theta functions must be well defined in a finite dimensional extension of p-adic numbers R_p, and their sum must converge so that p-adic norms of the summands must approach zero quickly. The outcome is a quantization of imaginary parts of moduli:

Im (Omega_ij)= (log(p)/pi)*n_ij.

The quantization is different for each prime p=2,3,5...

Also the imaginary parts of the moduli are quantized by number theoretic existence requirement, and there is a fascinating connection with the zeros of Zeta. The universal role of Rieman Zeta in the number theory suggests that the zeros of Riemann Zeta are universal in the sense that the partition functions 1/(1-p^(-z)} appearing in the product representation of Zeta exist in all p-adic number fields for zeros z= 1/2+iy of zeta.
This implies a sharpening of Riemann hypothesis: the numbers p^(n+iy), p any prime and y imaginary part of any zero of Zeta, exist in a finite-dimensional extension of R_q, q any prime. See


http://www.physics.helsinki.fi/~matpitka/tgd.html#Riema


If this highly non-trivial number theoretical conjecture, forming a corner stone for the realization of physics as a generalized number theory in the spirit of TGD, holds true, the quantized values of Teichmueller parameters are of form

Omega_ij= sum_i n_iy_i +n_ij,

y_i imaginary part of a zero of Zeta. For instance, for torus the ratio of side lengths and the angle between sides for the plane parallelogram defining it by the identification of opposite sides is quantized in terms of integers and zeros of Zeta.



5. p-Adic integral over moduli as a sum


p-Adic integration is not a well defined concept except in special cases (residy integral for which algebraic continuation applies, note the role of conformal invariance!). The solution of the problem is of course provided by the quantization of moduli. Integral over moduli becomes a sum converging extremely rapidly for values of p-adic prime which are physically
important. Only the very lowest terms are needed to calculate the contribution to the mass squared of elementary particle. The outcome is indeed the expected one and predicts correctly the mass ratios of e, mu, and tau identified in terms of g=0,1,2 topologies. Also quark masses and intermediate boson masses are predicted successfully with the above mentioned small prime k being the only free parameter.


6. Algebraic universality and dynamics

Since the sum (instead of integral) over moduli must result by an algebraic continuation from the real context, the quantization of moduli must occur also in the real context. The maxima of Kahler function of configuration space of 3-surfaces indeed naturally correspond to quantized moduli for partonic 2-surfaces X^2. The mere requirement that the maxima correspond
to values of moduli existing also as p-adic numbers fixes the quantization of moduli. Number theory would allow to solve the extremely non-linear dynamical problem of finding maxima for Kaehler function defined as an absolute minimum of Kahler action!


Matti Pitkanen


reader Lumo said...

Hi Matti! After you read the article, you asked whether it was a critique of LQG or M-theory? In that case, I would recommend you this:

http://www.nutritional-supplement-info.com/memory-supplements.html

But don't expect miracles.


reader Leucipo said...

Hmm, to give Feynman credit for reporting about polinesian "cargo cult" seems to me exagerate. At least, you could quote some nearer writer, form an instance Marvin Harris, who of course mentions these cults in some of his books.

This reminders me, that a pi^5 numerology example is sometimes attributed -in the network- to Feynman, sometimes to Lubos Motl...


reader Lumo said...

Hi Leucipo!

As far as I know, I was the first to see that the ratio of proton and electron masses is very close to 6.pi^5. Feel free to prove me wrong.

I know that Feynman did not *discover* the cargo cult, and I've provided you with a link to non-Feynman sources. ;-) On the other hand, Feynman is definitely the most famous person associated with the term - and especially with "cargo cult science" which *was* coined by Feynman, as far as I know.

My goal was not to talk about the social aspects of life in Polynesia, but about pseudoscience, and therefore Harris was totally irrelevant in my text, and of course I quoted Feynman. But you probably did not get what I wanted to say. Never mind.

Best
Lubos


reader Lumo said...

Let me copy the wikipedian definition of cargo cult science.

Cargo cult science
From Wikipedia, the free encyclopedia.

Cargo Cult Science is a term invented by Richard Feynman to describe a particular type of pseudoscience in which all the superficial aspects of scientific inquiry are adhered to, although the underlying causal link between the conditions and the outcome is not understood. Feynman introduced the phrase in a speech at Caltech in 1974, the transcript of which can be found in the book Surely You're Joking, Mr. Feynman! : Adventures of a Curious Character (Norton, 1985) and on many web sites. He based the phrase on an existing concept in anthropology, the cargo cult.
[edit]

http://clsdemo.caltech.edu/archive/00000051/02/CargoCult.htm


reader Plato said...

LubosHave you seen an actual hologram? It's a sticker that is completely two-dimensional, but when you look at it, it looks entirely three-dimensional. It's because of some interference going on in it, and we believe that something analogous takes place in quantum gravity. It's not just a shadow - it's a very fine picture with tiny strips whose wavelengths etc. encode the 3+1 dimensional information.Quickly, I thought of Moire Effect?


Is this view wrong?

Brane in this sense, was reduced from, Sunlight to shadow(geometricization of gravity to shadow on wall)?

I bet, I am sounding very confused and will partake of Matti supplements.:)

Maybe artists are cargo cultists, like Salvadore Dali, or literary people, like Hinton?


reader Leucipo said...

I missed the BBC link. Point conceded, partly at least. In revenge, let me to give a couple links that wander to attribute 6 pi^5 to Feynman or to random knowledge:

http://www.lacim.uqam.ca/~plouffe/610millions_english.doc.
http://groups.google.com/groups?hl=es&lr=&ie=UTF-8&selm=43opuv%24c5n%40sparcserver.lrz-muenchen.de
http://groups.google.com/groups?hl=es&lr=&ie=UTF-8&selm=325%40eildon.win-uk.net

And the third link gives a candidate publication
Wyler (Comptes Rendus) 20th Oct 1969 / 11 Jan 1971

And in this link, L.M is attributed the suggestion:
http://groups.google.com/groups?hl=es&lr=&ie=UTF-8&selm=24a23f36.0408072208.cf75b1d%40posting.google.com

Finally, LM himselfs claims credit here :-)
http://groups.google.com/groups?q=Lubos+6+pi+5&hl=es&lr=&ie=UTF-8&selm=Pine.LNX.4.31.0404092114490.17827-100000%40feynman.harvard.edu&rnum=2


reader Leucipo said...

But finally I found it... in the Physical Review!

Friedrich Lenz "The Ratio of Proton and Electron Masses" Phys. Rev. 82, 554 (1951)

A three-line article. I found it referenced in a long article from I J Good, whom sometimes is related with this formula.