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

Abstract

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 = 1/2mv 2 leads to Einstein's celebrated maximal energy equation E = mc 2 and how in turn this can be dissected into the ordinary energy density E(O) = mc 2 /22 and the dark energy density E(D) = mc 2 (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.

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

“…However what exactly is a Hausdorff dimension [39]? In nonlinear dynamics the word fractal dimension is used to mean more or less the same as the Hausdorff dimension [39].…”

“…However what exactly is a Hausdorff dimension [39]? In nonlinear dynamics the word fractal dimension is used to mean more or less the same as the Hausdorff dimension [39]. Consequently we may see the Hausdorff-fractal dimension not as a normal dimension but as a measure for the irregularity of a fractal shape, its ruggedness or smoothness.…”

“…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 [27] [35] [39] ( ) ( ) ( ) ( )…”

Computational Fractal Logic for Quantum Physics and Cosmology

“…In the light of such philosophy, number theory and some classical books such as that of Hardy [2] as well as the work of P. Erdos [17,18] and S. Ramanujan [19,20] became the new holy grail of nonlinear dynamics research [138][139][140][141][142]. In addition neither numerical experiments nor number theory were looked down upon, scorned or considered irrelevant to real physics in the relatively new and still vibrant field of deterministic chaos and fractals [137][138][139][140][141][142][143][144][145] . The Author is rather of the mildly strong opinion that this is a healthy scientific attitude worth exporting from nonlinear dynamics to high energy quantum physics and cosmology [48,141] and the next sections are concrete examples for what we mean by that.…”

“…iii) The 26 5 21 − = where 25 is the lower critical dimension of the classical Nambu-Goto strings [135][136][137] and the associated Gupta-Bleuler quantization [135,136] leading to the ghost-free condition a = 1, D = 26 and…”