Abstract :
We are going to show the link between the ε(∞) Cantorian space and the Hilbert spaces H(∞). In particular, El Naschie’s ε(∞) is a physical spacetime, i.e. an infinite dimensional fractal space, where time is spacialized and the transfinite nature manifests itself. El Naschie’s Cantorian spacetime is an arena where the physics laws appear at each scale in a self-similar way linked to the resolution of the act of observation. By contrast the Hilbert space H(∞) is a mathematical support, which describes the interaction between the observer and the dynamical system under measurement.
The present formulation, which is based on the non-classical Cantorian geometry and topology of spacetime, automatically solves the paradoxical outcome of the two-slit experiment and the so-called particle–wave duality. In particular, measurement (i.e. the observation) is equivalent to a projection of ε(∞) in the Hilbert space built on 3 + 1 Euclidean spacetime. Consequently, the wave–particle duality becomes a mere natural consequence of conducting an experiment in a spacetime with non-classical topological and geometrical structures, while observing and taking measurements in a classical smooth 3 + 1 Euclidean spacetime. In other words, the experimental fact that a wave–particle duality exists is an indirect confirmation of the existence of ε(∞) and a property of the quantum–classical interface.
Another direct consequence of the fact that real spacetime is infinite dimensional hierarchical ε(∞) is the existence of scaling law R(N), introduced by the author, which generalizes the Compton wavelength. It gives an answer to the problem of segregation of matter at different scales, and shows the role of fundamental constants such as the speed of light and Plank’s constant h in the fundamental lengths scale without invoking the principles of quantum mechanics.
