_{1}

We establish that ordinary energy, Casimir energy and dark energy are not only interlinked but are basically the same thing separated merely by scale and topology. Casimir energy is essentially a nano scale spacetime phenomenon produced by the boundary condition of the two Casimir plates constituting the Casimir experimental set up for measuring the Casimir force. By contrast dark energy is the result of the cosmic boundary condition, i.e. the boundary of the universe. This one sided Möbius-like boundary located at vast cosmic distance and was comparable only to the Hubble radius scales of the universe. All the Casimir energy spreads out until the majority of it reaches the vicinity of the edge of the cosmos. According to a famous theorem due to the Ukrainian-Israeli scientist I. Dvoretzky, almost 96% of the total energy will be concentrated at the boundary of the universe, too far away to be measured directly. The rest of the accumulated Casimir energy density is consequently the nearly 4% to 4.5%, the existence of which is confirmed by various sophisticated cosmic measurements and observations. When all is said and done, the work is essentially yet another confirmation of Witten’s T-duality and mirror symmetry bringing nano scale and Hubble scale together in an unexpected magical yet mathematically rigorous way.

The present paper is concerned with a wide range of problems connecting high energy quantum physics with relativity, quantum gravity and cosmology [

One of the many memorable words of wisdom that Prof. Sir Herman Bondi was known for is to the effect that once a great truth about nature is uncovered, it becomes so obvious and self evident that we are bound to wonder how we did not notice it for such a long time and how it could have eluded us all despite its compelling simple logic [

Referring to the present work the author would like to think that the thesis that we are about to reveal belongs in the same category which Prof. Bondi would have liked [

Before reading a single word or equation of the present paper, our advise to the prospective reader is to first have a long contemplative look at

In particular it can be shown that D(O) is the dimensionality of the zero set which models the pre-quantum particle while

where D = O is the Menger-Urysohn deductive topological dimension and

for the pre-quantum particle and

for the pre-quantum wave. Now using Fibonacci’s growth law and starting with D(O) using (a, b) = (0, 1) and D(1) (a, b) = (1, 0) as said, one obtains for positive dimension the following series

Remarkably our Fibonacci-like dimension series could be extended into the negative side using the same logic as follows [

Although the preceding computation is mechanistic and essentially elementary, it is quite tedious because of the heavy and slightly confusing notation for negative dimensions. It is at this point that one discovers the advantage of the equivalent notation used in E-infinity theory [

Here n is the Menger-Urysohn topological dimension while

and

correspondingly. For the core Hausdorff dimension of E-infinity we just need to set n = 4 and find [

We note for later use that for n = 2, which means the surface of the pre-quantum wave we find

Now we ask an important question, namely what happens when

By contrast for

This is clearly the end of our and any universe. Said in a mundane way, we basically have a Cantorian onion in the two dimensional projection as shown in

while the average dimension of the entire spacetime is given by

which is precisely the same value as that of the inversion of the cobordism of the empty set quantum wave. Thus we conclude that the picture of

Consider the classical Casimir set up and the hydromechanical analogy that goes with it. Bringing the two Casimir plates as near as a few nano units, we can consider the space between the plates practically totally empty of any “waves” or “particles” pushing them apart. Outside the plates spacetime resembles a stormy sea, which pushes the plates towards each other. Now the inside of the plates has the

The Casimir energy density is therefore given by the following ratio [

This is nothing but the familiar ordinary energy density of the universe that agrees with all previous analysis of the problem as well as accurate observations and measurements. Now this is only about 4.5 percent of the expected energy, so where is the rest is a natural question. However this question was answered some time ago. The bulk of the energy density is due to the quantum wave and is given by [

This is about 95.5 percent of the energy and is stored in the integer part of

The result is in complete agreement with that obtained using various methods. The fact that this 95.5 percent is located at the edge of the universe is explained within our five dimensional spacetime theory via Dvoretzky’s theorem [

There is a fascinating history behind the subject of this section testifying to the unity of science and showing that at the end, physicists and mathematicians of all moulds must think and work together on the most fundamental level to be able to push the borders of science forward [

Let me be more specific even at the risk of appearing elementary or even trivial. Let us start with Hardy’s quantum entanglement. The exact solution for two quantum particles entanglement is given by the well known accurate result

This is an implicit application of T-duality which is clear cut and in view of the experimental verification of Hardy’s entanglement and the theoretical success of M-theory points clearly to being a physical reality [

To put things into perspective we should not forget that we always referred to the Casimir effect and dark energy as different forms of quantum vacuum zero point fluctuation. Nevertheless the fact that

It consists of three main layers [

For a young man who said more than half a century ago “never trust anyone above thirty”, to become himself over seventy it is a time to pause and thank all those who really fundamentally influenced him and helped him in so many ways that he feels today grateful to them. On the top of the list his father, Major General Salah El Naschie (Alnashaee), then his teacher and friends, Prof. Dr. Alf Pflüger, Prof. T. Lehmann, Prof. E. Stein, Prof. R. Deboer, Prof. J.M.T. Thompson, Prof. T. Barta, Prof. Sir Herman Bondi, Prof. W. Martienssen, Prof. W. Greiner and Prof. IlyaPigogine. Last but not least, working with His Excellency Prof. Salah Al Athel of Saudi Arabia was one of the most important scientific experiences in the author’s life. There are of course many other names which the fading memory of an old man is not helping to remember but to all of them, I am truly indebted.

M. S.El Naschie, (2015) Hubble Scale Dark Energy Meets Nano Scale Casimir Energy and the Rational of Their T-Duality and Mirror Symmetry Equivalence. World Journal of Nano Science and Engineering,05,57-67. doi: 10.4236/wjnse.2015.53008