Cantorian space–time,Fantappie’s final group,accelerated universe and other consequences

In this paper we introduce Mohamed El Naschie’s ϵ^(∞) Cantorian space–time in connection with stochastic self-similar processes to give a possible explanation of the segregation of the Universe at fixed scale; then by considering the Fanntappie’s transformation group we show how the universe could appear accelerated on Cantorian space–time.     

Local scale invariance,Cantorian space–time and unified field theory

We develop field theoretic arguments for the unification of relativistic gravity with standard model interactions on El Naschie's Cantorian space–time. The work proceeds by showing the equivalence between the fundamental principle of local gauge invariance and the local scale invariance of space–time and matter fields undergoing critical behavior on high-energy scales. We focus on the transition boundary between the classical and non-classical regimes, the latter being characterized by generalized scaling laws with continuously varying exponents. Both relativistic gravity and standard model interactions emerge from the underlying geometry of Cantorian space–time near this transition boundary.     

Von Neumann Geometry and E(∞) Quantum Spacetime

In this paper, it is shown that von Neumann continuous geometry may be regarded as the first attempt towards formulating a general quantum spacetime geometry akin to that of Cantorian spacetime E^(∞) and noncommutative geometry.     

A Cantorian potential theory for describing dynamical systems on El Naschie’s space–time

In this paper we analyze classical systems, in which motion is not on a classical continuous path, but rather on a Cantorian one. Starting from El Naschie’s space–time we introduce a mathematical approach based on a potential to describe the interaction system-support. We study some relevant force fields on Cantorian space and analyze the differences with respect to the analogous case on a continuum in the context of Lagrangian formulation. Here we confirm the idea proposed by the first author in dynamical systems on El Naschie’s ϵ^(∞) Cantorian space–time that a Cantorian space could explain some relevant stochastic and quantum processes, if the space acts as an harmonic oscillating support, such as that found in Nature. This means that a quantum process could sometimes be explained as a classical one, but on a nondifferential and discontinuous support. We consider the validity of this point of view, that in principle could be more realistic, because it describes the real nature of matter and space. These do not exist in Euclidean space or curved Riemanian space–time, but in a Cantorian one. The consequence of this point of view could be extended in many fields such as biomathematics, structural engineering, physics, astronomy, biology and so on.     

Hilbert space,the number of Higgs particles and the quantum two-slit experiment

Rigorous mathematical formulation of quantum mechanics requires the introduction of a Hilbert space. By contrast, the Cantorian E-infinity approach to quantum physics was developed largely without any direct reference to the afore mentioned mathematical spaces. In the present work we present a novel reinterpretation of basic ε^(∞) Cantorian spacetime relations in terms of the Hilbert space of quantum mechanics. In this way, we gain a better understanding of the physical and mathematical structure of quantum spacetime. In particular we show that the two-slit experiment required a definite topology which is consistent with a certain fuzzy Kähler manifold or more generally a Cantorian spacetime manifold. Finally by determining the Euler class of this manifold, we can estimate the most likely number of Higgs particles which may be discovered.     

The Fine Structure Constant,Vacuum Polarization and the Geometry of e (∞) Space

This paper is a first attempt to derive the fine structure constant from the geometrical properties of the Cantorian manifold E ^(∞) which is assumed to model actual micro spacetime.     

The mean uncertainty and distance dimension of multidimensional cantorian spaces

Several ways of describing the internal structure of infinite point sets and determining a corresponding average dimension are outlined. Possible relevance to discretisation of quantum space-time are discussed. Finally, a method for determining the average internal distance between different Cantor points is proposed.The main conclusion is that a micro-scale Cantorian space-time may be based on an average Cantorian distance d_l⁽⁰⁾ ⋍ 0.629, while the smooth space-time is retrieved when d_l⁽⁰⁾ → 0.5, which means d_l⁽⁰⁾ tends towards the average distance of a continuous smooth line at a vanishing resolution. This correspond to the case when space is viewed from very far, at the macro-scale of classical physics.     

Self-similar and oscillating solutions of Einstein's equation and other relevant consequences of a stochastic self-similar and fractal Universe via El Naschie's ε(∞) Cantorian space–time

In this paper we make some suggestion regarding the unification of the fundamental forces and the age of the Universe in the context of the a stochastic self-similar and fractal Universe using El Naschie's ε^(∞) Cantorian space–time. We also show how Einstein's equation can admit for the scale factor a(t) a self-similar solution in agreement with our stochastic self-similar, fractal Universe and El Naschie's ε^(∞) Cantorian space–time. In addition, this solution is found to be oscillating one. Thanks to the first quantization it is possible to recast the equations in a Schrödinger-like form. Consequently, the presently observed large scale structure reflects the phenomenology of the Early Universe or of the microscopic world. Again it appears clear that the Universe and the structures inside must have a memory of its quantum origin as conjectured sometime ago.     

Hilbert,Fock and Cantorian spaces in the quantum two-slit gedanken experiment

On the one hand, a rigorous mathematical formulation of quantum mechanics requires the introduction of a Hilbert space and as we move to the second quantization, a Fock space. On the other hand, the Cantorian E-infinity approach to quantum physics was developed largely without any direct reference to the afore mentioned mathematical spaces. In the present work we utilize some novel reinterpretations of basic E^(∞) Cantorian spacetime relations in terms of the Hilbert space of quantum mechanics. Proceeding in this way, we gain a better understanding of the physico-mathematical structure of quantum spacetime which is at the heart of the paradoxical and non-intuitive outcome of the famous quantum two-slit gedanken experiment.     

Nuclear Spacetime Theories,Superstrings, MonsterGroup and Applications

The present paper is basically a collection of notes and remarks on various nuclear-quantum spacetime theories and their possible application in physics. Several subjects are considered including quantum gravity, fractional quantum Hall effect, knot theory, superstrings, noncommutative geometry, partially ordered sets, Cantorian spaces, branched polymers and sporadic 196884 dimensional monster group.     

Cantorian-fractal space-time and particles in a generalized magnetic field

In the fractal space-time theory the generalized precession, the generalized Stern–Gerlach experiment, the generalized Zeeman effect, the generalized Landau's levels and the generalized fractional quantum Hall effect are analyzed. The Cantorian structure of space-time implies the supplemental decreasing rate of the Ecliptic tilt, the “global” quantization in units of 36 km s⁻¹ and the filling factor ν=1/2n+1.     

On the Sporadic 196884-Dimensional Group,Strings and e (∞) Spacetime

Some arguments are presented using the dimension of the Golay code and the center density of intersecting spheres in 24-dimensional lattices to show that the 196884 dimensions of the so-called monster group can be regarded as hierarchical and reduced by clustering to a quasi expectation value of four, in analogy to the E ^(∞) spacetime dimensional reduction. The analysis suggests that the conjectured DNA-like Cantorian spacetime may resemble a giant error correction code.     

A Note on the Subtle Mean in Physics

It is shown that the subtle mean, which is the third power of the Golden number, has some quite interesting properties. These properties connecting diverse fields such as knot theory, subfactors, noncummutative geometry, Cantorian spacetime and quasi crystals are discussed and illustrated. It is conjectured that the subtle mean is the mean dimension of actual spacetime at the resolution of quantum physics.     

Deriving the curvature of fractal-Cantorian spacetime from first principles

The paper gives various exact derivations of the curvature of spacetime manifold at different energy scales within the frame work of a fractal-Cantorian theory. It is argued that at a Hausdorff spacetime dimensionality equal 4 + ³ = 4.236067977 the unification fractal spacetime Cantorian manifold possesses a curvature equal to K = 26 + k = 26.18033989.     

The Fibonacci code behind super strings and P-Branes. An answer to M. Kaku’s fundamental question

In “Beyond Einstein” the leading string theoretician and notable science writer Michio Kaku referred to what he labelled the ‘strange’ link between the E₈ exceptional Lie group and the various dimensionalities of strings and super string theories and commented on that by saying “If we could understand why the numbers 8, 10 and 26 continually crop up in super string theory, perhaps we could understand why the universe is four dimensional”.In the present work we demonstrate the existence of a Fibonacci code-like connection between the various coupling constants, charges and dimensionalities of super strings and P-Brane theories. This code is based on the Fibonacci numbers and the golden mean and in the final analysis, may be attributed to the deterministically chaotic nature of the hyperbolic Cantorian sets fixing the geometry and topology of quantum spacetime.     

Fantappiè's group as an extension of special relativity on ε Cantorian space–time

In this paper, we will analyze the Fantappiè group and its properties in connection with Cantorian space–time. Our attention will be focused on the possibility of extending special relativity. The cosmological consequences of such extension appear relevant, since thanks to the Fantappiè group, the model of the Big Bang and that of stationary state become compatible. In particular, if we abandon the idea of the existence of only one time gauge, since we do not see the whole Universe but only a projection, the two models become compatible. In the end we will see the effects of the projective fractal geometry also on the galactic and extra-galactic dynamics.     

On the Cantorian Structure of Time in Relativity

Identifying space with a superconducting cosmic dust and associating the physicaltime with the complex diffusion time, one can show that time has a Cantorian structure. In thiscontext, the Hubble effect may be understood as a paired diffusion in Cantorian space.     

Another weak forms of faint continuity

A function f:X→Y is said to be faintly continuous [Kyungpook Math. J. 22 (1982) 7] if f⁻¹(V) is open in X for every θ-open set V of Y. In this paper, we introduce and investigate two weaken forms of faint continuity which are called faint α-continuity and faint γ-continuity. We obtain their characterizations, their basic properties and their relationships with other types of functions between topological spaces also, some results in (A.A. El-Atik, A study of some types of mappings on topological spaces, M.Sc. Thesis, Tanta University, Egypt, 1997) are improved. We speculate that weak-faint continuity may be relevant to the physics of fractal and Cantorian spacetime.