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Based on Witten’s T-duality and mirror symmetry we show, following earlier work, the fundamental complimentarity of the Casimir energy and dark energy. Such a conclusion opens new vistas in cold fusion technology in the wider sense of the word which we tackle via fractal nano technologies leading to some design proposals for a nano Casimir-dark energy reactor.

Modern theoretical physics has a truly marvellous story to tell about its development [^{2} is in fact the sum of two quantum parts [^{2}/22 and that of the quantum wave energy E(D) = mc^{2} (21/22) so that Einstein did indeed hit the nail on its head, quantumly speaking [

The main two elements or building blocks of E-infinity diagrams is the zero set and the empty set [

(a) The zero set D(H) = ϕ

(b) The empty set D(H) = ϕ^{2}

where D(H) is the Hausdorff dimension and^{2} in symbolic diagram (see

From the diagrams of

1) When the distance between the two plates of ^{3} while the average spacetime density is basically the fractal five dimensional average

not related in any way to the Riemann curvature of spacetime but to the chaotic fractality of spacetime in full agreement with our picture which we adopted based on Feynman’s conjecture that gravity is similar to van der Waals fluctuation of micro spacetime frequently termed Feynman-El Naschie van der Waals quantum gravity conjecture [

2) Now we look at the other extreme where the distance between the Casimir plates is equal to the diameter of our universe as shown schematically in ^{3} which is working in the opposite direction so that the net repulsive topological pressure pushing the boundary of the universe outwardly is given by

empty set albeit in five dimensional Kaluza-Klein space [

We start from our exact picture of quantum spacetime [^{2} and ϕ^{3} are not only Hausdorff dimensions but they can also be understood as a topological frequency or critical parameters of corresponding limit cycles. Therefore we can use the well known comparison theorem of eigenvalue due to Dunkerley and figure the combined critical value or joint Hausdorff dimension as follows:

Consequently we have

where

Second it is clear that 11 + ϕ^{5} is the isomorphic length of the Penrose fractal universe and therefore 22 + k = (2) (11 + ϕ^{5}) is the corresponding diameter of the universe [

This is the dimensionality 11 of Witten’s M-theory plus a Hardy quantum entanglement part, namely ϕ^{5}.

Similar to the quasi probability of Wigner’s quantum mechanics in phase space [

The internal logic of our approach rivals that of Weyl-Wigner quantum mechanics in phase space and its logical underpinning by Groenewold and Moyal [^{n} is getting increasingly small, it is clear that this means the “Cantor set” is becoming thinner and tending to real nothingness or the absolute empty set [

For want of better words, ϕ^{3} was called with equal justification universal fluctuation [

In E-infinity the topological speed of light is a velocity charge being between zero and infinity in a circular world interval where the golden mean is a topological average. Thus the topological energy is [

This is Hardy’s topological energy. To get “real” energy we use Newton [

This is thus what we call mass in the quantum small world. In the large cosmic world on the other hand, mass is

Consequently we have

From the above the complimentarity of E(O) and E(D) follow:

Similar considerations apply to the Mageuijo-Smolin formula for quantum gravity energy.

To obtain E(O) and E(D) and visa versa we have the following transformation [

The same applies to a fractal de Sitter universe [

That means ordinary energy is

and dark energy is

Let us start with Heisenberg’s uncertainty [^{3} as ^{2} as its derivative are fixed numbers which can be written in decimal expansion in a finite way nor the quasi phase space spanned by ϕ and ϕ^{2}. The same is true for the product ϕ^{3} which is ambiguous in a fundamental way because ϕ could be seen as a distance or a mean velocity while ϕ^{2} could be interpreted as acceleration or mean velocity squared. However for the union of the two sets, the zero set ϕ and the empty set ϕ^{2} we have the remarkable collapse in a simple integer [

It would be a mistake to think this result is trivial. However, written in symbolic form

or

when compared to its totally chaotic decimal expansion [

and find on the top of that a well known fact, namely that ϕ is the Hausdorff dimension of a random triadic Cantor set.

At the risk of appearing facetious, we would like to seriously propose that in no minor measure a draw back of E-infinity Cantorian spacetime theory is that it is excessively simple. A theory should be simple but not excessively so. Being excessively simple puts a theory at risk of being called trivial as an easy shot by members of the “voluntary opposition”. In his early days working in applied mechanics, the author was in awe vis-à-vis the work of a great scholar Prof. Cliff Trusdell who coined the word rational mechanics [

In the following we show a simple deep connection between a few fundamental aspects of E-infinity and Weyl-Wigner theory [

a) If we take ϕ to mean a coordinate then ϕ^{2} is the quasi derivative and

b) Seen that way, then

c) Since the VAK is a Hamiltonian “strange” attractors conjectured by Rene Thom to represent the equilibrium states of quantum mechanics [

We presented in this relatively short paper a general theory for quantum spacetime and zero point energy fluctuation based on E-infinity and related mathematical concepts. Our main results and conclusions may be summarized in the following rather important points [

1) Casimir energy and ordinary energy density of the cosmos are not only identical conceptually but identical numerically.

2) Casimir energy and cosmic dark energy are complimentary in the most strict mathematical and physical meaning.

3) The difference between Casimir energy and dark energy is a difference of boundary condition where the boundary of the holographic boundary of the universe is a one sided Möbius-like manifold (see

4) Using a heap of space filling fractal nano spheres, we can build in principle a mini universe and use it as an energy reactor (see

5) The main conclusion is a natural consequence of mirror symmetry and Witten’s T-duality (see Figures 2-4 of Ref. [

The author is truly indebted to the work and many discussions with the outstanding French scholar Prof. Jean-Paul Auffray. He is equally indebted to the work of Prof. L. Hardy as well as the work and advice of Nobel Laureate G. ‘tHooft. Last but not least the work of Prof. J. Mageuijo, Prof. Lee Smolin, Prof. J. Ambjorn and Prof. R. Loll were extremely important.

M. S. ElNaschie, (2015) Quantum Fractals and the Casimir-Dark Energy Duality—The Road to a Clean Quantum Energy Nano Reactor. Journal of Modern Physics,06,1321-1333. doi: 10.4236/jmp.2015.69137