The role of dissipation in the ‘t Hooft-El Naschie-Ord quantum systems

9th December, 2010.
E-infinity communication No. 47
The role of dissipation in the ‘t Hooft-El Naschie-Ord quantum systems
Almost all realistic engineering systems have friction losses or any other kind of dissipation. As a structural engineer and applied mechanics scientist El Naschie dealt as long ago as 1976 with nonconservative, dissipative mechanical systems. In his paper in ‘Solid Mechanics Archives, Vol. 4, August 1979 (published by Stijthoff & Noordhoff Int. Publishers, Holland) he developed a finite element-like method (finite element is the engineer’s version of Regge calculus of general relativity). The method he employed, invented by Belgian engineer van den Dungen (Bull. Acad. Ray Belg. Sci Ser 1945 (31), pp. 659-668) consists of joining two dissipative systems, one with energy losses and another with energy gain (a so called flutter set), balancing each other and thus forming a conservative Hamiltonian system. In a paper published in 1995 in CS&F entitled “A note on quantum mechanics, diffusional interference and information” he extended his two dissipative systems forming one conservative system idea to the Schrödinger equation by two conjugate complex Schrödinger equations, one going forwards and the other going backwards in time (see CS&F, 5(5), pp. 881-884 (1995)). The importance of this paper was immediately recognized by Prof. G. Ord whose model is essentially very similar although it may not seem to be that way without careful examination. It all boiled down to the need for an additional negative sign which will become apparent later on. It could be that the Nobel Laureate became quite interested in nonlinear dynamics which always includes dissipation and a dissipative ‘strange attractor’ after meeting El Naschie on several occasions around the year 2000 including a conference in Riyadh and another in Cairo. It is clear that ‘t Hooft must have been thinking about these things for some time before that because he realized that the notion of time in relativity is fundamentally different from that in quantum mechanics and because he had a controversy with Steven Hawkings about the information paradox of black holes. Ultimately ‘t Hooft wrote several papers connecting the loss of information with dissipation and applied that to a new quantum mechanics which he called deterministic quantum mechanics. Similar to E-infinity ‘t Hooft used fluid turbulence as a paradigm for his theory. However at that time ‘t Hooft new nothing about quasi attractors in Hamiltonian systems because the VAK (the vague attractor of Kolmogorov) was not yet recognized by Mohamed El Naschie and thus not yet incorporated into his work. The VAK first conjectured as the stationary states of quantum mechanics by French topologist and the inventor of catastrophe theory, Rene Thom was considered much later (see for instance the paper “Strange non-dissipative and non-chaotic attractors and Palmer’s deterministic quantum mechanics” by G. Iovane and S. Nada published in CS&F, 42, pp 641-642 (2009).

Part II of communication No. 47
It is important to realize that although the VAK has no physical friction to give it stability, it has a mathematical substitute for the lack of friction, namely the irrationality of the winding number. Noting that the golden mean is the most irrational number, it follows that a golden mean winding number is the most stable orbit for a dynamic system and that is the reason why the mass of the elementary particles which could realistically be observed experimentally is always a function of the golden mean and its power. This also follows from von Neumann-Connes’ dimensional function. Dimensions are related to the coupling constant and these are in turn related to energy and thus the mass of elementary particles. In 2007 M. El Naschie gave a Lagrangian formulation to his basic 1979 idea in a paper entitled “On gauge invariance, dissipative quantum mechanics and self-adjoint sets. The paper was dedicated to his by that time very close friend Gerard ‘t Hooft in celebration of his 60th birthday (see CS&F, 32 (2007), pp. 271-273). In the meantime Ord refined his work by introducing the ani-Bernulli diffusion mimicking quantum mechanics (see for instance Annals of Physics, 324 (2009), pp. 1211-1218) where he refers as usual to the very same 1995 paper of El Naschie. The fact however is that all these different formulations are basically different faces of the same multi-dimensional coin. We can obtain the needed ‘negative’ sign by considering an adjoint flutter set with energy gain or a dissipative set with energy losses. We can also change a Bernulli random walk using zero and plus one to an anti-Bernulli random walk with 0, +1 and ̶ 1. We can introduce anti-commuting Grassmanian coordinates a proposed by ‘t Hooft and used in super string theory or we can go back to fundamentals and consider all the empty sets with their negative Menger-Urysohn dimensions as done by Mohamed El Naschie
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