Cantorian space–time and Hilbert space: Part II—Relevant consequences

In this paper, we will show the consequences of the link between ε(∞) and H(∞). Starting from El Naschie’s ε(∞) nature shows itself as an arena where the laws of physics appear at each scale in a self–similar way, linked to the resolution of the observations; while Hilbert’s space H(∞) is the mathematical support to describe the interaction between the observer and dynamical systems. The present formulation of space–time, based on the non-classical, Cantorian geometry and topology of the space–time, automatically solves the paradoxical outcome of the two-slit experiment and duality. The experimental fact that a wave-particle duality exists is an indirect confirmation of the existence of ε(∞). Another direct consequence of the fact that real space–time is the infinite dimensional hierarchical ε(∞) is the existence of the scaling law R(N). The present author proposed it as a generalization of the Compton wavelength. This rule gives an answer to segregation of matter at different scales; it shows the role of fundamental constants like the speed of light and Plank’s constant h in the fundamental lengths scale without invoking the methodology of quantum mechanics. In addition, we consider the genesis of E-Infinity. A Cantorian potential theory can be formulated to take into account the geometry and topology of ε(∞) in the context of gravitational theories. Consequently, we arrive at the result of the existence of gravitational channels.