replace in our mind quantum waves and quantum spacetime by the mental picture of real sea waves. Imagining two large walls brought just a meter apart and placed in relatively shallow water, then clearly the two walls are more likely to be pushed together by the outside waves because the water between the walls is obviously much calmer than in the open se water outside. In other words, the Casimir effect is not a mystery. The real mystery is in the quantum wave and quantum spacetime is the solution. In Figure 1we present a gedanken experiment to show that the net Casimir topological force is equal to half the universal global E-infinity quantum spacetime fluctuation, i.e. 32φwhere ()151φ=is the Hausdorff dimension of the zero set modelling the quantum particle. Rather than cluttering Figure 1of the schematic Casimir set up with many labels and notations we de-cided to leave it bare and explain it meticulously in this section. The reader should, in the back of his mind, number the above three perfectly conducting but uncharged Casimir plates in Figure 1as one to three, from left to right. At the two extremities the topological force-pressure is identified here with the Hausdorff dimension of
M. S. El. Naschie 215 the zero set, i.e.φof the quantum particle. Inside however, i.e. between plates 1 and 2 as well as 2 and 3, we have the empty set with the Hausdorff dimension 2φ, which models the quantum wave. Consequently, follow-ing this “topological” conception there is a net external topological force acting from outside on the inside equal to 23φφφ−=. This is not the force per single plate but the intrinsic fluctuation pressure 3φ, which is the in-verse of the Hausdorff dimension of the E-infinity quantum spacetime in which the entire gedanken experiment is taking place and where the spacetime topological dimension is exactly 4 while the corresponding well known value of the Hausdorff dimension is 33144.23606799φφ= +=. Seen that way the Casimir topological force is given by ()32110kφ=+where ()331kφφ=−. This corresponds to a dimensional value of ()()()32Casimir2Pc dφ=where cis the speed of light and is the Planck constant. This is rather close to the approximate value found by A. Zee for the same arrangement assuming a massless scalar field rather than an electromagnetic field, namely ()()2π24cdas given on page 66 of his classical text book “Quantum Field Theory In A Nutshell” . Let us stress once more a basic topological insight of this gedanken experiment with physical deep meaning. If φrepresents the particle aspect of the quantum while 2φis the corresponding re-presentation of the wave aspect, then by intersection theory ()()23φφφ=represents both simultaneously. This means physical Casimir energy is a spacetime pure topological energy. Since the Davies-Unruh temperature is nothing but a dynamical Casimir effect then by the same token it is an effect produced by the random fractal to-
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- Winter '16