The stationary states of quantum mechanics and the golden mean in E-infinity

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 (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.
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