_{1}

The work gives a natural explanation for the ordinary and dark energy density of the cosmos based on conventional quantum mechanical considerations which dates back as far as the early days of the quantum theory and specifically the work of Max Planck who seems to be the first to propose the possibility of a half quanta corresponding to the ground state, i.e. the energy zero point of the vacuum. Combining these old insights with the relatively new results of Hardy’s quantum entanglement and Witten’s topological quantum field theory as well as the fractal version of M-theory, we find a remarkably simple general theory for dark energy and the Casimir effect.

The true nature and origin of dividing energy into two main categories namely ordinary energy which we are able to measure and dark energy which should be there but could not be found or measured in any direct way is one, if not the most puzzling questions of modern science [

The present work sprang out of such a realization and our final result and explanation of Casimir energy [

We know very well, at least since J. von Neumann’s pointless continuous geometry [

where

The innocent conclusion of the above half quanta is that our postulate gained mainly from experiments that quanta are indivisible cannot be as straight forward as one could naively have thought and who knows, it may open the door to unsuspected connections related to fractional-Hall effects and similar things [

As far as the present Author is concerned there are few modern results in quantum physics that can rival Hardy’s magnificent gedanken experiment regarding the maximal quantum entanglement probability for two quantum particles [

where

Quantum field theory is primarily concerned with investigating the topological invariants of a theory and is the result of pioneering efforts of Schwartz, Attaya, Donaldson and Witten [

We all have a pretty reasonable understanding and intuitive feel for what a topological dimension means. However what exactly is a Hausdorff dimension [

where the length of the unit interval within which the random Cantor set lives is unity. That way we see that

which is what we find in our classical world where we are dealing with almost infinitely many particles and that is why in classical mechanics we do not have measurable entanglement of any kind. From the preceding discussion we see clearly that we could replace

Let us now synthesize and fuse together the preceding result and discussion into a single coherent unity. We start with stating the final result. This is first that the vacuum zero point energy is found from replacing

Second this energy is clearly the cause behind the Casimir effect which is observed via a change of the boundary condition created by the two uncharged but conducting Casimir plates brought at nano distance of each other [

This clearly means that E_{o} is in this case equivalent to the ordinary energy density of the cosmos E(O):

Consequently it follows that the dark energy density is simply [

Comparing these results with the actual cosmic measurements of WMAP, Planck and type 1a supernova [

The result that ^{2}, from it [^{2} using ironically quantum mechanics which Einstein was not able to bring himself to embrace without many reservations to say the least [^{2} from the above. First we have to admit that E = mc^{2} is already included in_{o} was not found by appealing to any spacetime. It is simply the vacuum energy density so that to find the entire energy density of our spacetime it should be multiplied with the topological “volume” of our spacetime [_{o} with the Hausdorff dimension of our spacetime and expect to find a reasonable answer. However what is the Hausdorff dimension of our universe? One could be tempted to answer hastily that it is our

This is twice the dimension of the fractal version of Witten’s M-theory. Proceeding this way one finds [

The second possibility is far more straight forward and is nothing more than adding E_{o} = E(O) and E(D) together and finding that [

Either way we see that E = mc^{2} consists of two quasi quantum components well hidden inside the deceptively simple Einstein’s beauty E = mc^{2} [^{2} not only into two parts E(O) and E(D) but into three parts making a distinction between dark matter energy E(DM) and pure dark energy D(DE) where

while

In the case of writing E in three parts, we cannot escape the coupling term

where

This coupling could be taken to be approximately

exactly as should be.

We gave a derivation for the ordinary energy density and the dark energy density of the universe starting from and based upon conventional and generally accepted quantum mechanical principles. In particular we relied upon a fact introduced probably for the first time by Max Planck, which shows that even in the absence of any real photon, completely empty spacetime has a non-zero energy. From there we went on to show that using this half quanta of Planck which is in the meantime part of most text books on quantum mechanics, we can explain not only the Casimir effect but could also explain the division of energy into ordinary measurable energy as well as dark energy which we cannot measure directly. Thus unlike our previous publications, we did not need to invoke new advanced mathematics nor really any new concepts beyond what one is taught in an advanced course or two in a good university undergraduate program in physics.

Mohamed S. El Naschie, (2016) Max Planck Half Quanta as a Natural Explanation for Ordinary and Dark Energy of the Cosmos. Journal of Modern Physics,07,1420-1428. doi: 10.4236/jmp.2016.712129