From a Dual Einstein-Kaluza Spacetime to ‘tHooft Renormalon and the Reality of Accelerated Cosmic Expansion

We use a dual Einstein-Kaluza spacetime to calculate the exact energy density of dark energy and dark matter using a novel topological computation method. Starting from the said spacetime and ‘tHooft’s topological renormalon as well as the corresponding symmetry group, we show how the zero set quantum particle and the empty set quantum wave interact with the vacuum and give rise to pure dark energy and pure dark matter all along with ordinary energy density of the cosmos. The consistency of the exact calculation and the accurate observations attests to the reality of ‘tHooft’s renormalon dark matter, pure dark energy and accelerated cosmic expansion.

Einstein’s Dark Energy via Similarity Equivalence, ‘tHooft Dimensional Regularization and Lie Symmetry Groups

Realizing the physical reality of ‘tHooft’s self similar and dimensionaly regularized fractal-like spacetime as well as being inspired by a note worthy anecdote involving the great mathematician of Alexandria, Pythagoras and the larger than life man of theoretical physics Einstein, we utilize some deep mathematical connections between equivalence classes of equivalence relations and E-infinity theory quotient space. We started from the basic principles of self similarity which came to prominence in science with the advent of the modern theory of nonlinear dynamical systems, deterministic chaos and fractals. This fundamental logico-mathematical thread related to partially ordered sets is then applied to show how the classical Newton’s kinetic energy E = 1/2mv2 leads to Einstein’s celebrated maximal energy equation E = mc2 and how in turn this can be dissected into the ordinary energy density E(O) = mc2/22 and the dark energy density E(D) = mc2(21/22) of the cosmos where m is the mass;v is the velocity and c is the speed of light. The important role of the exceptional Lie symmetry groups and ‘tHooft-Veltman-Wilson dimensional regularization in fractal spacetime played in the above is also highlighted. The author hopes that the unusual character of the analysis and presentation of the present work may be taken in a positive vein as seriously attempting to propose a different and new way of doing theoretical physics by treating number theory, set theory, group theory, experimental physics as well as conventional theoretical physics on the same footing and letting all these diverse tools lead us to the answer of fundamental questions without fear of being labelled in one way or another.

A Combined Heterotic String and Kähler Manifold Elucidation of Ordinary Energy,Dark Matter,Olbers’s Paradox and Pure Dark Energy Density of the Cosmos

We utilize the topological-geometrical structure imposed by the Heterotic superstring theory on spacetime in conjunction with the K3 Kähler manifold to explain the mysterious nature of dark matter and its coupling to the pure dark energy density of the cosmos. The analogous situations in the case of a Kerr black hole as well as the redundant components of the Riemannian tensor are pointed out and the final result was found to be in complete agreement with all previous theoretical ones as well as all recent accurate measurements and cosmic observations. We conclude by commenting briefly on the Cantorian model of Zitterbewegung and the connection between Olbers’s paradox and dark energy.

On a Fractal Version of Witten’s M-Theory

Starting from Witten’s eleven dimensional M-theory, the present work develops in an analogous way a corresponding dimensional fractal version where . Subsequently, the new fractal formalism is utilized to determine the measured ordinary energy density of the cosmos which turns out to be intimately linked to the new theory’s fractal dimension via non-integer irrational Lorentzian-like factor: where is Hardy’s probability of quantum entanglement. Consequently, the energy density is found from a limiting classical kinetic energy to be Here, is ‘tHooft’s renormalon of dimensional regularization. The immediate logical, mathematical and physical implication of this result is that the dark energy density of the cosmos must be in astounding agreement with cosmic measurements and observations.

The Counterintuitive Increase of Information Due to Extra Spacetime Dimensions of a Black Hole and Dvoretzky’s Theorem

As per Hawking and Bekenstein’s work on black holes, information resides on the surface and there is a limit on it amounting to a bit for every Planck area. It would seem therefore that extra dimensions would logically lead to a hyper-surface for a black hole and consequently a reduction of the corresponding information density due to the dilution effect of these additional dimensions. The present paper argues that the counterintuitive opposite of the above is what should be expected. This surprising result is a consequence of a well known theorem on measure concentration due to I. Dvoretzky.

The Emergence of Spacetime from the Quantum in Three Steps

The paper presents a very simple and straight forward yet pure mathematical derivation of the structure of actual spacetime from quantum set theory. This is achieved by utilizing elements of the topological theory of cobordism and the Menger-Urysohn dimensional theory in conjunction with von Neumann-Connes dimensional function of Klein-Penrose modular holographic boundary of the E8E8 exceptional Lie group bulk of our universe. The final result is a lucid sharp mental picture, namely that the quantum wave is an empty set representing the surface, i.e. boundary of the zero set quantum particle and in turn quantum spacetime is simply the boundary or the surface of the quantum wave empty set. The essential difference of the quantum wave and quantum spacetime is that the wave is a simple empty set while spacetime is a multi-fractal type of infinitely many empty sets with increasing degrees of emptiness.

Quantum Dark Energy from the Hyperbolic Transfinite Cantorian Geometry of the Cosmos

The quintessence of hyperbolic geometry is transferred to a transfinite Cantorian-fractal setting in the present work. Starting from the building block of E-infinity Cantorian spacetime theory, namely a quantum pre-particle zero set as a core and a quantum pre-wave empty set as cobordism or surface of the core, we connect the interaction of two such self similar units to a compact four dimensional manifold and a corresponding holographic boundary akin to the compactified Klein modular curve with SL(2,7) symmetry. Based on this model in conjunction with a 4D compact hy- perbolic manifold M(4) and the associated general theory, the so obtained ordinary and dark en- ergy density of the cosmos is found to be in complete agreement with previous analysis as well as cosmic measurements and observations such as WMAP and Type 1a supernova.

Spacetime from Zitterbewegung

Quantum particles are assumed to have a path constituting a random fluctuation super imposed on a classical one resulting in a golden mean spiral propagating in spacetime. Consequently, the dimension of the path of the quantum particle is given by one plus the random Cantor set Zitterbewegung, i.e. 1+Øwhere Øis the golden mean Hausdorff dimension of a random Cantor set. Proceeding in this way, we can derive the basic topological invariants of the corresponding spacetime which turned out to be that of E-infinity spacetime 4+Ø3 as well as a fractal Witten’s M-theory 11+Ø5. Setting Ø3 and Ø5 equal zero, we retrieve Einstein’s spacetime and Witten’s M-theory spacetime respectively where Ø3 is the latent Casimir topological pressure of spacetime and Ø5 is Hardy’s quantum entanglement of the same.

Time Dimension and Ordinary Cosmic Energy Density Are Fractal Effects

In a short, neat and credible analysis, it is established that time is a fractal effect of the Cantor set-like topology of micro spacetime. This effect as well as the ordinary cosmic energy density of the universe is shown to be a direct consequence of Hardy’s probability of quantum entanglement. Finally and as a general conclusion, we point out the importance of understanding the fractal origin of time as well as spacetime for resolving certain types of paradoxes arising in quantum information science.

Max Planck Half Quanta as a Natural Explanation for Ordinary and Dark Energy of the Cosmos

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.

Application of Dvoretzky’s Theorem of Measure Concentration in Physics and Cosmology

Using Dvoretzky’s theorem in conjunction with Bohm’s picture of a quantum particle inside a guiding quantum wave akin to De Broglie-Bohm pilot wave we derive Einstein’s famous formula E = mc2 as the sum of two parts E(O) = mc2/22 of the quantum particle and E(D) = m c2 (21/22) of the quantum wave where m is the mass, c is the speed of light and E is the energy. In addition we look at the problem of black holes information in the presence of extra dimensions where it seems initially that extra dimensions would logically lead to a hyper-surface for a black hole and consequently a reduction of the corresponding information density due to the dilution effect of these additional dimensions. The present paper argues that the counterintuitive opposite of the above is what should be expected. Again this surprising result is a consequence of the same well known theorem on measure concentration due to I. Dvoretzky. We conclude that there are only two real applications of the theorem and we expect that many more applications in physics and cosmology will be found in due course.

From E=mc^(2) to E=mc^(2)/22—A Short Account of the Most Famous Equation in Physics and Its Hidden Quantum Entanglement Origin

Einstein’s energy mass formula is shown to consist of two basically quantum components E(O) = mc2/22 and E(D) = mc2(21/22). We give various arguments and derivations to expose the quantum entanglement physics residing inside a deceptively simple expression E = mc2. The true surprising aspect of the present work is however the realization that all the involved “physics” in deriving the new quantum dissection of Einstein’s famous formula of special relativity is actually a pure mathematical necessity anchored in the phenomena of volume concentration of convex manifold in high dimensional quasi Banach spaces. Only an endophysical experiment encompassing the entire universe such as COBE, WMAP, Planck and supernova analysis could have discovered dark energy and our present dissection of Einstein’s marvelous formula.