- 10th December, 2010. E-infinity communication No. 49. The stationary states of quantum mechanics and the golden mean in E-infinity
- 10th December, 2010. E-infinity Communication No. 50. The placebo effect in mainstream physics
- 10th December, 2010. E-infinity communication No. 51. The quantum sets of D. Finkelstein, E-infinity sets and the paper of D. Benedetti
Interesting parts are green. My comments are [bracketed red].
10th December, 2010.
E-infinity communication No. 49
The stationary states of quantum mechanics and the golden mean in E-infinity
The stationary point in classical mechanics is well known. A complete classification was given long ago in the work of Andrenov which was frequently used by M.S. El Naschie and J.M.T. Thompson in their work on stability theory, theory of bifurcation and René Thom’s catastrophe theory. For details see El Naschie’s book “Stress, stability and chaos in structural engineering – An energy approach”, McGraw Hill, London (1990) as well as Thompson’s “Instabilities and castrophies in science and engineering”, John Wiley, Chichester (1982). However something was overlooked for which Sir James Lighthill [may he rest in peace] (who had a very high regard for M.S. El Naschie and who helped to establish Chaos, Solitons & Fractals) had to apologize publicly to the public at large in an article published in the Proceedings of the Royal Society. Sir James said that by overlooking the generic chaotic nature of classical mechanics and letting so many people wrongly believe for such a long time that classical mechanics is generically deterministic, the community of theoretical and applied mechanics, of which he was the Head at the time (IUTAM President), misled the educated society at large. Five major discoveries were made through the revolution of deterministic chaos.
First the work of E. Lorenz established the one additional missing attractor of classical mechanics, namely the so called strange attractor. The great German chaos scientist Otto Rössler simplified the attractor of Lorenz. Rössler’s attractor is besides the Lorenz attractor, one of the main paradigms of chaos. Second the discovery of universalities in chaos by Mitchell Feigenbaum which extended original earlier work by S. Grossmann. Third the mathematical theory of turbulence and strange attractors of D. Ruelle and F. Takens. Fourth period 3 implies chaos by J. York who found that a single Cantor set is the backbone of all complex strange attractors and chaotic behavior and finally B. Mandelbrot who gave the correct geometry of deterministic chaos. This geometry is fractals. Again fractals is what the great G. Cantor saw with his inner eyes as the geometry of his Cantor set and the related gallery of monsters as H. Poincaré described them. All these strange attractors belong to dissipative dynamics. However what about Hamiltonian systems? These systems are more relevant to quantum mechanics. Seeing a connection between chaos in Hamiltonian systems and quantum mechanics was the achievement of René Thom and applying it to high energy physics and the mass spectrum of elementary particles was the achievement of Mohamed El Naschie and E-infinity theory. Making a great deal of bad jokes and running obscene blogs about non-science is the anti-achievement of those who do not tire from wasting their lives on defamatory allegations against E-infinity and those working in our group. At a minimum this shows the low self esteem of these internet characters who are so thick skinned that they force themselves on society just because the internet tolerates almost anything and any ignoramus.
In what follows we give briefly the most important points about the VAK in quantum mechanics and E-infinity theory:
1. The KAM nested concentric tori complex picture which Poincaré was reluctant to draw in 1899 is exactly that which R Thom proposed in 1975 as the Hamiltonian analogy of the strange attractor in differential dynamics. It was later on named the VAK. Like all fractals the VAK is almost self-similar. If one small VAK inside a large VAK is enlarged, we find another VAK in it and so on like a solenoid. In infinite dimensions as in E-infinity theory the VAK vague stability is due to the irrationality of the KAM orbits and may be used according to R. Thom as a model for the stable states of quantum mechanics.
2. Influenced by the thinking of René Thom, Mohamed El Naschie used to say in his numerous lectures which he gave in the last twenty years in Germany, Italy, Spain and particularly Egypt and Saudi Arabia “Descartes explained everything using his vortices and hooked atoms but could calculate nothing; Newton calculated everything using his inverse square law but could explain nothing. Only a geometrical theory like Einstein’s theory, string theory or E-infinity theory could explain and compute almost everything.
3. The vague attractor was studied by Kolmogorov, Moser and Arnold and that is why it is called KAM or VAK. The golden mean is easily shown to be to the most irrational number because in a continued fraction we have only the smallest non-zero integer, namely one. Consequently the golden mean is the worst irrational number which could be approximated by a rational number and that explains the stability of any periodic orbit with the golden mean winding number against perturbation.
4. The infinite complexity of geometrical form is reminiscent of the paradoxical notion of quantum field theory where the energy density of the vacuum is infinite.
5. The onset of turbulence is characterized by the replacement of a vague attractor of a Hamiltonian dynamics with finite-dimensional pseudo group of symmetries by a large ergodic set similar to a Cantor set. Thus we must distinguish two types of catastrophe point, the ordinary catastrophe and the second type of open, chaotic set with the complicated topology of a Cantor set. This second type is what is the case for noncommutative geometry and E-infinity theory.
6. We emphasize one more the wide spread misconception that Cantor sets and fractals appear only at the Planck length and Planck energy. The work of people as well established as ‘t Hooft and Lee Smolin as well as young otherwise excellent researchers such as Dario Benedetti all suffer from this misconception. Fractals have very little to do with Wheeler and Hawking’s foam and are not bound to the Planck energy. They appear generically even in classical systems at low energy.
7. As the level of the VAK there is no essential difference between quantum phase space, a Hilbert space or a fractal Cantorian spacetime. The excellent work of T.N. Palmer needlessly suffers from this misconception which is based on a widespread prejudice related to wrongly connecting fractals to the so called Wheeler-Hawking spacetime foam.
10th December, 2010.
E-infinity Communication No. 50
The placebo effect in mainstream physics
The violent effect of the establishment against new ideas is well documented and historically well researched. E-infinity has experience just such a violent effect only on the internet which gives perverted maniacs the same voice as anyone and may even be hired by sinister forces to silence competition. This is definitely the same with certain blogs hired in the north of Germany as well as a small neighbor country to scandalize as far as India and China. This possibility did not exist at the time of Einstein’s relativity and his Nazi opponents, some of whom were Nobel Laureates. Seen in this way the hired, as distinct from the real scientific opposition against E-infinity is harmless compared to that against Einstein’s relativity (see for instance the article in Physics World, April 2003).
The main reason in modern times for the rejection of a theory and unfair competition is funding money. String theoreticians may be excused for thinking that the book of Peter Woit “Not even wrong” in which he attacks string theory on all levels may have been written on the instigation of the leading scientist working in loop quantum gravity. We venture no opinion on this particular case but we think that as far as the young, innocent, unsuspecting scientists go, the problem may be related to a form of placebo effect. You expect that a theory or a scientist is bad because your supervisor told you so, then you find it bad. On the contrary, if you are told by your supervisor, or thesis advisor or Editor-in-Chief of a journal that a theory or a scientist is pure genius, then you will find it really pure genius. This is exactly the placebo effect in scientific research. Please think about it.
10th December, 2010.
E-infinity communication No. 51
The quantum sets of D. Finkelstein, E-infinity sets and the paper of D. Benedetti
Prof. David Ritz Finkelstein is one of the earliest pioneers of quantum set and the construction of spacetime in a way similar to deriving thermodynamics from the motion of atoms. [I met Finkelstein three decades ago but he wouldn't remember me. He looks like a regular guy now but at the time he presented the appearance of a crazed Gandalf.] This is the original idea of El Naschie’s work which he acknowledges to have taken over from Finkelstein and Wizecker. Finkelstein is a deep thinker who was described in one of the acclaimed books of Lee Smolin as one of the seers in science today. The monad of Finkelstein space is the null set. The null here is not the dimension and we can consider his null set to be our null set as well as our empty set. We recall that E-infinity follows Suslin set theory and constructs everything from a Suslin tree starting with the empty set, the null set and the unity set. In E-infinity these are ( ̶ 1, ϕ2), (0, ϕ) and (1, 1). We note that in E-infinity we have Mandelbrot-El Naschie [o_O El Naschie often names things after himself, as has been noted] notion of the degrees of emptiness of an empty set leading to the totally empty set ( ̶ ∞, 0). The monad of E-infinity is random Cantor sets in general. Monad in the terminology of ‘t Hooft are called the building blocks of spacetime. ‘t Hooft does not work with sets. However a scientist somewhat close to him and very close to Prof. R. Loll, namely Prof. Fay Dowker works with partially ordered sets as explained in earlier communications. The work by D. Finkelstein was developed considerably by Heinrich Saller (see Quantum space-time-gravity by J. Baugh, D. Finkelstein, M. Shiri-Garakani and H. Saller). A particularly good summary of D. Finkelstein’s pioneering work is “Quantum sets and Clifford algebras”, Int. J. of Theoretical Physics, Vol. 21, No. 6/7 (1982). In the abstract of this paper Finkelstein says “… Quantum set theory may be applied to a quantum time space and quantum automaton.” In the introduction of his lecture in the presence of Feynman and Wheeler he said “several of us here including Feynman, Fredkin, Kantor, Moussouris, Perti, Wheeler and Zuse suggest that the universe may be discrete rather than continuous and more like a digital than analog computer. C.C. von Weizaecker (the student of Heisenberg and the elder brother of the past President of the Federal Republic of Germany) worked this path since the early 1950’s and we have recently benefitted from the relevant work of J. Ford (Ford was an expert in deterministic chaos and together with two of El Naschie’s friends, J. Casati and B. Cherekov pioneered the science of quantum chaos which is sometimes confused with the theory of Cantorian-fractal spacetime).
In the rest of this communication we address the technical aspect of E-infinity and show how to derive all the results of the work of D. Dario Benedetti from first principles in a far simpler and transparent manner using E-infinity. It is difficult to do this without writing equations but we will try our best.
First we consider our Cantorian E-infinity spacetime to be the union of infinitely many elementary monads Cantor sets. A single random Cantor set has, by the theorem of Mauldin-Williams, a Hausdorff dimension equal to the golden mean. The higher order Cantor sets will have a Hausdorff dimension equal ϕ to the power of n where n is 0, 1, 2… Adding all these dimensions together one finds the finite dimension of the large Cantorian space. Thus from summing from zero to infinity of ϕ to the power n, one finds 1 + ϕ + ϕ2 + ϕ3 + … and so on to infinity. The sum is exactly equal 2 + ϕ. This corresponds in the Connes-El Naschie [-_-] dimensional function or bijection to a Menger-Urysohn dimension of exactly 3. However this has not been gauged in terms of the original monad ϕ. Therefore we have to divide 2 + ϕ by ϕ and this gives us the famous dimension 4.23606799 which corresponds exactly to the Menger-Urysohn topological dimension equal to 4. To show that 2 + ϕ corresponds to the topological dimension of 3 we just exclude the non-fractal dimension 1 from our summing. That means we start from n = 1 to n = ∞. That way the total dimensions become 2 + ϕ ̶ 1 = 1 + ϕ. To obtain the gauged dimension we divide by ϕ and find 2 + ϕ again which means the topological dimension is 3. Thus we have obtained from first principles the 4 and 3 dimensions of spacetime and space only with a single assumption which is that the union of all the elementary monads represents our spacetime. The monads themselves are formed by intersections of two and more monads. Unlike other theories our monads are explicit and well defined. They are random Cantor sets. The zero set for instance is given by two dimensions. First the topologically invariant Menger-Urysohn dimension is zero and second the Hausdorff-fractal dimension which is in this case ϕ. Thus the zero set is fixed by (0, ϕ). Using a simple elementary cobordism argument for (0, ϕ), the neighborhood or border is a wave given by the empty set ( ̶ 1, ϕ2). We could go on that way indefinitely. That means we have ( ̶ 2, ϕ3), ( ̶ 3, ϕ4) and finally (0, ̶ ∞) as mentioned earlier on. In fact we can determine the world sheet of fractal string theory. We did that when we summed from n = 1 to n = ∞ and found the total Hausdorff dimension to be 2 + ϕ ̶ 1 = 1 + ϕ. This corresponds in the Connes-El Naschie dimensional function bijection to a two dimensional object. The fractal world sheet is topologically two dimensions but Hausdorffly more than one dimensional and less than two dimensions. It is a fractal area-like with a Hausdorff fractal dimension equal 1 + ϕ = 1.618033989.
In fundamental scientific research there are two directions. We either keep improving and even patching and fixing our older theories or we start afresh from different or slightly different basic assumptions and principles. E-infinity is the second possibility. Dario Benedetti is the usual first possibility. Einstein’s relativity and quantum mechanics as well as string theory and loop quantum gravity started as the second possibility like E-infinity. In fact ‘t Hooft’s quantum field theory started exactly as E-infinity theory. Now string theory and quantum field theory are the establishment and the improvements and patching are characteristic for these important developments. In real life we need both methods and both philosophies. When we reach the same conclusions, this is then the promised land.