_{1}

^{*}

Dark energy is shown to be the absolute value of the negative kinetic energy of the halo-like quantum wave modeled mathematically by the empty set in a five dimensional Kaluza-Klein (K-K) spacetime. Ordinary or position energy of the particle on the other hand is the dual of dark energy and is contained in the dynamic of the quantum particle modeled by the zero set in the same five dimensional K-K spacetime. The sum of both dark energy of the wave and the ordinary energy of the particle is exactly equal to the energy given by the well known formula of Einstein *E=mc*^{2 }which is set in a four dimensional spacetime. Various interpretations of the results are presented and discussed based on the three fundamental energy density equations developed. In particular where E is the energy, m is the mass and c is the speed of light, is Hardy’s quantum entanglement and gives results in complete agreement with the cosmological measurements of WMAP and Supernova. On the other hand gives an intuitive explanation of negative gravity and the observed increased rate of cosmic expansion. Adding *E *(ordinary) to *E *(dark) one finds which as we mentioned above is Einstein’s famous relativity formula. We conclude that similar to the fact that the quantum wave interpreted generally as probability wave which is devoid of ordinary energy decides upon the location of a quantum particle, it also exerts a negative gravity effect on the cosmic scale of our clopen, i.e. closed and open universe. Analysis and conclusions are framed in a reader friendly manner in **Figures 1-14** with detailed commentary.

We utilize the wave-particle duality [1,2] and quantum entanglement [2-8] as fundamental features of quantum mechanics [1,8] as well as classical field, quantum set theory [

where E is the energy, m is the mass, c is the speed of light, is Hardy’s quantum entanglement and [12,17,18] (see

about percent of the total energy of the cosmos [18,19] (see Figures 1-12). In turn the quantum particle is modeled by the zero set represented dimensionally by von Neumann-Connes dimensional function [8,44] of a Penrose quasicrystal fractal tiling space [21-23]. The broken mirror image of the quantum particle is the halo-like quantum probability wave which is the source of a negative energy anti-gravity (see Figures 9 and 11) stemming from modeling the quantum wave via the empty set of the same Penrose quasicrystals fractal tiling [21-23,30-34,44]. Subsequently the energy of the quantum wave was found to be [12,18]

It was then found to the Author’s delight that the sum of ordinary energy of the particle and absolute value of the negative dark energy of the wave added together is exactly the energy predicted by Einstein’s formula [2,3, 17]

Our most important conclusion in this work is that Einstein’s gives the correct total energy formula for an infinitely large Cantor set-like clopen i.e. closed and open universe [24-29] but it does not distinguish between ordinary energy and negative dark energy of a basically fractal cosmos. Noting that measurement leads to quantum wave collapse, it is completely normal that we cannot measure dark energy using present day technology. In the following sections we will attempt to explain more accurately but concisely the points transacted above. For a quick overview, the entire analysis and results are summarized in Figures 1-3, while valuable details and additional information are presented in Figures 4-12. In Figures 13 and 14 we give a naive yet very instructive geometrical visualization and interpretation of the wave-particle duality in five dimensional Kaluza-Klein spacetime.

Ordinary energy of the universe is directly proportional to the fractal nature of spacetime [4-12,30-34]. More accurately it is the multiplicative hyper Hausdorff measure or volume of the zero set i.e. the quantum particle in a five dimensional Kaluza-Klein universe. The zero set which possessrs the cardinality of the continuum is fixed using the Bi-dimensions formula [

i.e. the topological dimension and the Hausdorff dimension so that we find

where as in von Neumann-Connes’ non-commutative geometry [8,22,33] and its well known dimen dimensional function when applied to Penrose fractal clopen i.e. closed and open universe tiling [21,44].

Dark energy on the other hand is given by the dual additive hyper Hausdorff measure or volume of the empty set i.e. quantum wave in Kaluza-Klein spacetime [4,6]. Thus we start from which is the empty set given by [

i.e. [8,44].

For the hyper Hausdorff volume of the fractal zero set of ordinary energy [

In a sense this Hausdorff measure is the “volume” or the energy density of a quantum particle [

The volume could be naively interpreted geometrically as the generalization of the volume of a cube to a pseudo 5D cube with each side being of a length equal to. (See Figures 13 and 14)

On the other hand for the additive Hausdorff “volume” which is equal the length of the empty set [22,33] of dark energy we have the addition or union rule of sets in 5 K-K dimension and one finds (see Figures 1(a) and (b))

We used here the union of sets rather than the intersection because of the duality between particle and wave [2,8] (see Figures 13 and 14). Consequently the Hausdorff measure of the empty set in K-K spacetime represents the energy density of a quantum wave which does not exist in the usual conventional quantum interpretation because a conventional quantum wave in orthodox quantum mechanics is a probability wave and has no

ordinary energy and collapses at measurement [1,2]. and collapse at measurement [1,2]. Again a possible naïvegeometrical interpretation of is being a one dimensional “surface” i.e. the circumference of a pentagon with each of the 5 sides being of a length equal to (see

From the above points (3) and (4) we see that the total geometrical density or pseudo Hausdorff “volume” is the sum of both the fractal part of spacetime and the voids “empty” part of spacetime. This means (see Figures 1(a) and (b) as well as Figures 13 and 14)

This is an extremely important result because an arithmetic average geometrical density or Hausdorff “dimensional volume” would give us

On the other hand the measure or “volume” or geometrical density of the unitarity set is and leads directly to Einstein’s celebrated energy density formula [2,5,17] (see Figures 1 and 2)

where E is the energy, m is the mass and c is the speed of light [2,3,12,17].

The fractal meaning of the constancy of the speed of light is explained in

Noting the result of point (2.5) we see that Einstein’s equation may be viewed as based on an arithmetic average density or “volume” of 5D Kaluza-Klein space which does not differentiate between fractals due to voids and the voids in spacetime themselves and consequently Einstein’s energy density is simply given by the famous equation based on continuous geometry and unit hyper volume equal to one [12,26]:

On the other hand ignoring the density of the voids and considering only the fractal part of spacetime represented by the five dimensional zero set, we then find the quantum relativity formula of ordinary energy density validated by direct cosmic measurement namely [12,17, 18] (see

Note that division by 2 in the above equation is due to arithmetic averaging and not due to using Newton’s kinetic energy because in this particular interpretation velocity tends to be the velocity of light without multiplying it with as in unit interval physics using Sigalotti’s critical value [28,29]. For an alternative derivation of see Figures 3-8 as well as Figures 13 and 14.

By contrast if we consider spacetime voids only as represented by the quantum wave i.e. the empty set in 5D Kaluza-Klein theory (which means that it is the so-called dark energy of the cosmos), then we have the following complementary quantum relativity energy density formula: (see

where

[22,34]

From points discussed in the preceding sections we can conclude that Einstein’s formula is blind to the difference between ordinary energy i.e. quantum particle energy and dark energy i.e. the quantum wave energy because it presupposes a continuous smooth spacetime and never addresses any questions related to fractality or sparseness of spacetime which is the cause of the constancy of the speed of light at the classical interface being a fractal average (see

If we consider the effect of the two topological dimensions of the zero set and the empty set on our average pseudo Hausdorff Kaluza-Klein hyper volume or geometrical density then we could work out the following modified negative arithmetic average [

Consequently dark energy acquires in this case the

negative sign of the topological dimension of the empty set corresponding to the basic compactification equation of bosonic string theory [2,6,17]

This negative sign (see

We conclude that while fractals i.e. the zero set or phenomenologically the quantum particle is behind ordinary energy, voids in spacetime corresponding to the empty set i.e. the quantum wave i.e. the halo of the quantum particle [8,23,33,44] are not only behind dark energy but also are the cause of negative gravity and the increased rate of cosmic expansion [9-11,15,16]. Negative gravity is essentially due to negative curvature of spacetime. If we see gravity as the Riemannian curvature of spacetime, then negative curvature [

Ordinary energy density and dark energy density are two dual yet complementary faces to a one common energy given by the sum of both which is equal to Einstein’s famous equation [

Here m is the five dimensional Kaluza-Klein mass [38,39], c is the speed of light and is the Planck energy. Using the unit interval set theoretical value for it was reasoned earlier on that we must have: and inserting in E we find that [12,17] (see

exactly as should be. However using Witten’s T-duality [2,4,6,40] (see

and inserting in E one finds

where [22,33].

Thus we have demonstrated in the present work and presumably for the first time that dark energy is essentially the absolute value of the negative energy density of the Hawking-Hartle quantum wave solution of Feynman-Wheeler-DeWitt Schrödinger equation of the entire universe and as such it induces the hitherto unknown phenomena of negative gravity which we and others consider to be the cause of the observed increased rate of cosmic expansion [9-18]. This physical interpretation makes a great deal of sense because the set theoretical definition of the quantum wave is the empty set represented by whose topological dimension is minus one (−1). Consequently the accurate way to write down the quantum wave energy density is [8,9,32]

where 5 is the dimension of Kaluza-Klein spacetime [42, 43]. To gain an even deeper understanding of the broken symmetry and duality between particle and wave and consequently between ordinary energy and negative or dark energy, let us start this time from Newton’s classical expression for the kinetic energy [

Recalling the set theoretical duality between the quantum particle zero set and the quantum wave empty set [

and (19)

as well as the dual transformation given for particle and wave as in the following box:

Box No. 1. Particle-wave duality transformation.

For particle

For wave

Then inserting in one finds the following particle and wave energy as per the next box:

Box No.2. Quantum particle ordinary energy and quantum wave dark energy.

For particle

For wave

Let us contemplate the meaning of in and 5 in 5m as well as in. The simplest way to explain it in the present context is in. This stems from Sigalotti’s critical speed given by The on the other hand is where is the Hausdorff dimension of a four dimensional set given by and representing the core of fractal-Cantorian spacetime [21,22]. At the same time is the global quantum entanglement probability or what some call the counter factual effect of the spacetime surrounding a quantum particle [

In this case would mislead us into wrongly thinking that dark energy is equal to Einstein’s energy (see Figures 1 and 4)

This is initially a surprising result but it is a misunderstanding as well because it shows that E(Einstein) is obtainable from a fractal five-dimension Kaluza-Klein spacetime while dark energy is the child of a mere fivedimensional non-fractal Kaluza-Klein spacetime. However the difference is namely that is the factor responsible for the additional energy which is nothing but the ordinary energy which when added to non-fractal 5D Klein Kaluza [38,39] i.e. to dark energy gives the total energy, namely that of Einstein’s energy. This confirms all of our earlier results namely that could be obtained via either a smooth 4D spacetime or via a delicate analysis using a fractal five dimension spacetime (see

In other words we can either interpret in as a hyper volume or Hausdorff measure and division by 2 as averaging or we see it as the transformation and and division by 2 as coming from using Newton’s kinetic energy.

We conclude this section by stressing that there are three and not only one Kaluza-Klein theories. The first is the classical 5D theory of Kaluza [

Dark energy is a non-classical form of negative energy density stemming from the quantum wave empty set characteristic and we found it quantitatively to be given by the absolute value of

where could be interpreted as the local Hardy’s type quantum entanglement of two particles or the Hausdorff dimension of the empty set and.

Ordinary energy on the other hand is given by the positive energy of the zero set describing the quantum particle and found in the present work to be equal [

where is the famous golden mean Hardy’s quantum entanglement [2,31].

There are numerous other methods and theories which lead to essentially the same conclusion regarding ordinary positive energy density being equal to Einstein’s energy divided by 22 which are summarized in Figures 5-8. In particular

A particular tantalizing result is that the sum of both energies, dark energy and ordinary energy, is a neat confirmation of Einstein’s famous relativity formula because

The negatives of the dark energy on the other hand confirm earlier string theoretical conjectures that negative gravity is due to negative energy of the compactified 22 dimensions which produce negative curvature as per the bosonic string dimensional equation [2,4,12]

Following the same reasoning one can write the factor of dark energy as

which proves that of ordinary energy and of dark energy have opposite signs in any event. In fact the present theory strongly suggests the physical reality of Kaluza-Klein extra fifth dimension [42,43] as well as the fractality of spacetime and ontological reality of the quantum wave [8,13,14,20,43,44] all apart from the reality of dark energy which we cannot see or measure because measurement collapses the quantum wave.

For a different view of negative gravity via general relativity and comparison with Heterotic string theory see

It is worth noting that negative dimensions are behind the experimentally observed negative Kelvin absolute temperature [

We could understand the repelling effect of dark energy on a cosmic scale as a phenomenon basically due to the fact that the motion of quantum objects and its spatial location is determined completely by the quantum wave following prescription [1-3]. It is a matter of speculation that the spinning of Gödel’s universe (see