Schrödinger’s cat meets Arnold’s cat in El Naschie’s infinity garden

E-infinity communication No. 44

Schrödinger’s cat meets Arnold’s cat in El Naschie’s infinity garden.

No one knows exactly why E. Schrödinger the Austrian Nobel Laureate in physics and founder of wave mechanics chose a cat to subject to a most cruel type of quantum measurement. The reasons for the Russian V. Arnold, probably one of the greatest mathematicians of all time to deform a cat’s picture beyond recognition using his well known map are even less clear. In all events both cats are alive, well and jumping in E-infinity’s golden garden. The two eigenvalues of Arnold’s cat map happened to be the inverse of the golden mean (1/ϕ) = 1 + ϕ and the inverse of the golden mean squared (1/ϕ)2 = 2 + ϕ. Now adding both values or multiplying them gives exactly the very same result, namely (1 + ϕ) + (2 + ϕ) = 4.23606799 and (1 + ϕ) (2 + ϕ) = 4.23606799. In other words union and intersection of the two transfinite sets gives the same result. One can easily see that the first set have the topological dimension of an area just like a world sheet in string theory. This area has a fractal dimension equal 1 + ϕ = 1.61833989. The second set on the other hand is topologically three dimensional, however its Haudorff dimension is 2 + ϕ = 2.168033989. Topologically the union of the two sets is 2 + 3 = 5 dimensional. However seen from the Hausdorff dimension view point, the two sets give a dimension equal 4 + ϕ3 = 4.23606799 which corresponds to exactly 4 and only 4 topological dimensions. In other words one topological dimension is hidden. The 5 dimensions are used only for embedding the 4.23606799 fractal dimension. On the other hand we know very well that 2 + ϕ = 2.61803389 corresponds exactly to 3 topological dimensions of the tangible world. The only thing left for interpretation is that 1 + ϕ which is topologically 2 dimensional is that these two dimensions stand for time and the spin ½ fermionic dimension or alternatively for the fifth Kaluza-Klien compactified or the cyclic dimension of electromagnetism.
The reader is referred to the following paper for simple geometrical visualization of the Russian doll-like E-infinity space with 4.23606799 Hausdorff dimension corresponding to exactly only 4 topological Menger-Urysohn dimensions. See for instance “An irreducibly simple derivation of the Hausdorff dimension of spacetime”, Chaos, Solitons & Fractals, 41, (2009), pp. 1902-1904, particularly Table 1 on page 1903. See also “The theory of Cantorian spacetime and high energy particle physics (an informal review)”, Chaos, Solitons & Fractals, 41 (2009), pp. 2635-2646, in particular Fig. 1 and Fig. 2 on pages 2636 and 2639 respectively.
It remains to say that it is the fine structure of fractals contributing 0.23606799 = ϕ3 where ϕ is the golden mean = 0.618033989 which causes the equality of union and intersection of the Cantor sets spanning E-infinity spacetime and leading to the infinite but hierarchal dimensionality of this fractal Cantorian spacetime manifold which is the cause of the persistent illusion that a quantum object on a fractal particle can be said to be in two different spacetime fractal ‘points’ at the very same fractal ‘time’. Looked upon it from a distance, this intricate non-smooth and chaotic Cantorian-fractal spacetime appears smooth with only four spacetime topological dimensions. To show this in the most clear quantative way we just need to approximate the irrational number to the simplest rational number. That means 2 + ϕ = 2.618033 will be 2.5 = 5/2 and 1 + ϕ = 1.618033 will be 1.5 = 3/2. Adding together we find 2.5 + 1.5 = 4. On the other hand multiplication leads to a different dimension known from the theory of dimensional regularization, namely (5/2)(3/2) = 3.75 = 4 ̶ 0.25. This corresponds in E-infinity to 4 ̶ k where k = 0.18033989 of the transfinite Heterotic super string theory. It is vital to recall the importance of Arnold’s cat map in the study of quantum chaos as well as the discussion of the role of irrational numbers in combined harmonic oscillators by Prof. G. ‘t Hooft in his paper mentioned in an earlier communication “The mathematical basis for deterministic quantum mechanics”. In fact on page 7 of the arXiv paper he writes “….If the frequencies have an irrational ratio (which in the terminology of El Naschie and nonlinear dynamics means irrational winding number), the period of the classical system is infinite and so a continuous spectrum would be expected.” Sooner or later we are confident that those working in quantum field theory and have come as far as this will realize that K-theory, noncommutative geometry and E-infinity is the right way to come to what Nobel Laureate ‘t Hooft hopes to find, namely some kind of deterministic quantum mechanics.
E-infinity Group.