Dynamical systems on El Naschie’s ε(∞) Cantorian space–time

In this paper we analyze classical and quantum systems, whose motion is not on classical continuous path, but on a Cantorian one. The results is that an harmonic material support with an external force is equivalent, under some assumptions, to a classical or quantum harmonic system on a continuous support. We make some suggestions 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. In particular, we study some relevant force field on Cantorian space and analyze the differences respect to the analogous case on continuum, from classical and quantum point of view. The idea that we want stress in this paper is that a Cantorian space could explain some relevant stochastic and quantum processes, if the space acts as an harmonic oscillating support, such as happen in Nature. This means that a quantum process, in some cases, could be explained as a classical one, but on a non continuous and fractal material support. We consider the validity of this point of view, that in principle could be more realistic, since it describe the real nature of the matter and space, which does not only exist in Euclidean space or curved one, but in a Cantorian one. 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 Schrodinger-like form. Consequently, the presently observed large scale structure reflects the phenomenology of the Early Universe or of the microscopic world.