For Anosov systems try these:

http://arxiv.org/pdf/1009.2127.pdf

http://www.math.u-psud.fr/~anantharaman/ICMP2.pdf

http://www.lptl.jussieu.fr/user/lesne/MSCS-entropy.pdf

Thank you, Demetris, for these links. I will take a look. Do these links discuss fine stucture constant? Best wishes

Actually, what Adler tried to do in the 1972 was to see, whether the value of the fine structure constant *at atomic scales*, i.e. e^2/4pi epsilon_0 hbar c)~1/137... could be obtained by some physical mechanism, not yet known-and the answer, apparently, was negative, given what we know, in the sense that additional assumptions are required. This doesn't have anything to do with Shannon information entropy, however and I think that it's not at all clear why such a relation could exist.

However, we do know that the coupling constants of a physical system are determined by flow equations, so we are led to the study of these equations and the space they define. This leads us to the study of the corresponding dynamical systems. But mentioning ``Shannon information entropy'' in this context is wrong, because the dynamical systems of the flow equations are the objects of interest.

The equations must come from a physical model, in the case at hand the Standard Model. We do not possess a quantitative understanding of the coupling constant flows of the Standard Model, that would allow us to address the question, however, in this way. We only know the flow equations within perturbation theory here.

What we can do, since 1974 and the work of Wilson, to overcome the limitations imposed by perturbation theory, is to study the flow equations derived from a lattice regularization of the field theories (what's called the Monte Carlo Renormalization Group). We do not have sufficient quantitative control to address the question here, either, yet, however.

Thank you, Stam, for your answer. I do not mean that Adler,s paper discussing shannon entropy. I cite it because it discusses possible physics behind the fine structure constant.

In fact, until now i am not aware of a single paper discussing link between shannon entropy and fine structure constant. Best wishes

Dear Demetris, the papers you cited do not discuss fine structure constant, albeit the first paper discussing Anosov systems. Thank you anyway for your answer

Victor, This paper might be of interest: "Axiomatic approach to the cosmological constant" Christian Beck Physica A 388:3384-3390 (2009) arXiv:0810.0752v2 Abstract: A theory of the cosmological constant Lambda is currently out of reach. Still, one can start from a set of axioms that describe the most desirable properties a cosmological constant should have. This can be seen in certain analogy to the Khinchin axioms in information theory, which fix the most desirable properties an information measure should have and that ultimately lead to the Shannon entropy as the fundamental information measure on which statistical mechanics is based. Here we formulate a set of axioms for the cosmological constant in close analogy to the Khinchin axioms, formally replacing the dependency of the information measure on probabilities of events by a dependency of the cosmological constant on the fundamental constants of nature. Evaluating this set of axioms one finally arrives at a formula for the cosmological constant that is given by Lambda = (G^2/hbar^4) (m_e/alpha_el)^6, where G is the gravitational constant, m_e is the electron mass, and alpha_el is the low energy limit of the fine structure constant. This formula is in perfect agreement with current WMAP data. Our approach gives physical meaning to the Eddington-Dirac large number hypothesis and suggests that the observed value of the cosmological constant is not at all unnatural.

Thank you, Michael, for your suggestion. Yes i know that paper by Christian Beck, but it discusses a theory of lambda, instead of structure constant. Thank you anyway. Best wishes

More info about Anosov:

http://www.mathnet.ru/php/person.phtml?personid=8772&option_lang=eng

I think you can find there the seminar that you have lost.

Probably you want to translate this book:

Anosov, D. V., Aranson, S. Kh., Bronshtein, I. u., and Grines, V. Z.

Smooth Dynamical Systems, VINITI Series on Contemporary Problems in

Mathematics. Fundamental Directions, vol. 1, VINITI, Moscow, 1985, pp. 151-242 (in Russian).

Welcome Victor, If you follow Beck's analogy then this paper of mine might also help, citing Beck and more generally connected with the fine structure constant:

"Classical Quintessence and the Cosmological Constant." Quintessence is also found in the paper on "Wolfgang Pauli and the Fine-Structure Constant"

Demeteris, thank you for the url. Yes Anosov is from Steklov Math. Institute, but i don,t plan to translate his works. Best wishes

Dear Michael, thank you for your answer. If you have a paper discussing fine stucture constant, surely i would appreciate if you would like to share it. Where is the url of your paper? Thanks.

Perhaps this morning i will share what i can recall from Anosov lecture about fine stucture constant. Best wishes

Dear all, here is my short article discussing possible relation between fine structure constant and Shannon entropy. This is what I can recall from Anosov's lecture in my memory, since my note on his lecture is lost. Hopefully this article will stimulate further discussion. best wishes

Hi Victor, Check the references for more https://www.researchgate.net/publication/215969837_Wolfgang_Pauli_and_the_Fine-Structure_Constant

Article Wolfgang Pauli and the Fine-Structure Constant

Dear Michael, thank you for your paper. Yes it seems interesting to find links to platonic geometry or other mathematical objects. But if we follow the John Wheeler's phrase It is from Bit, then it seems that we should find explanation of physics from entropy. Of course, it does not mean that somebody has found the final words on this subject of fine structure constant.

Ps: i read somewhere a paper about Pauli's mental depression and his connection with Jung. Do you know that paper? Best wishes

Yes Victor, Jung's analysis of Pauli's dreams and visions were part of the impetus for the fine-structure constant paper. An interesting observation was that the artwork of the Emerald Tablet of Hermes is in the same chapter as Pauli's World Clock analysis. So if you want to connect the roots of entropy to alchemy, Carl Jung and Wolfgang Pauli were definitely on the pathway to that physical discovery.

@Victor, at your formula:

a=p/S

If we take k=1000 in S=k*ln(2) then we obtain

a=0.00072134...

which is the 1/10 of a=1/137.

(Of course we can take k=1387 and then a=0.000520077, which has nothing to do with real value of a).

So, what is your point?

Thank you, Demetri, for your comment. My point is that there is simple formula which can connect fine structure constant and shannon entropy, although it depends on k variable. But that is what i can recall from memory, i may make a mistake. Btw, i think many other proposed formulas also include free parameter. Best wishes

You must first establish *why* the Shannon entropy of a completely unrelated system could be relevant for giving the value of the fine structure constant at any scale. Absent such an argument-which I don't see where it could come from, I'm at a loss to understand what is the point of the calculation.

Regarding the formulae: In Shannon's expression, k is Boltzmann's constant, which you can, of course, set to the *value* you like, since it has *dimensions*-but that implies that your relation alpha =p/S cannot make sense, since alpha is dimensionless, so the ratio cannot be identified with the fine structure constant, under any circumstances-you don't alter dimensions by a change of units.

Dear Stam, thank you for your comment. Yes you are right that alpha should be dimensionless; perhaps we shall introduce other parameter there. Anyway i only try to recover from my memory the lecture given by Prof. Anosov in 2008, i just remember that also explained about casimir effect. But i am not sure how to include the casimir effect into the formula.

As a precaution, this is not intended to be an exact proof, it is written to stimulate further thinking on possible relation between fundamental constants. Best wishes

Michael, have you read a dissertation about Wolfgang Pauli in Jungian perspective? The dissertation is written by John Richard Gustavson, available at http://arxiv.org/pdf/1003.3223.pdf. Best wishes

Yes Victor, Have you seen this one?

"Fine Structure Constant α ~ 1/137.036 and Blackbody Radiation Constant αR ~ 1/157.555" Ke Xiao http://vixra.org/abs/1101.0037

The fine structure constant α = e2/hc ~ 1/137.036 and the blackbody radiation constant αR = e2(aR/k4B)1/3 ~ 1/157.555 are linked by prime numbers. The blackbody radiation constant is a new method to measure the fine structure constant. It also links the fine structure constant to the Boltzmann constant.

Dear Michael, thank you for the reference to blackbody radiation constant. Sure i will read it. Best wishes