Fractal space signatures in quantum physics and cosmology - I. Space, time, matter, fields and gravitation

The present paper explores the physical structure of two fundamental concep ts, spacetime and fractals, both of which suffer from a lack of a satisfact ory definition. We carefully consider the essential properties of these con cepts and attempt to answer the questions: Have space and time an independe nt existence (with respect to matter and fields) and can we speak of a frac tal space or a fractal time? How can we translate into a physical model the geometrically (formal) picture of a fractal? Since spacetime represents the arena in which the rest of physics unfolds i t is of interest to determine at how many different levels fractal spacetim e can make its influence felt. We distinguish four levels of fractalization . A real intrinsic fractalization of space may arise on the first (quantum gravity) and the fourth (cosmic) level where the texture of space and stron g nonlinear physics of space are involved. We show that even if the mathematical fractal curves are nondifferentiable (or finite piece-wise smooth curves in the case of real fractals), we can s till study their properties applying the Finsler theory of sprays which app roximates physical fractals and may have the same general properties as ide al fractals (self-similarity, self-affinity, etc). This point of view is fu rther supported by generalising the concepts of allometry and hierarchy of possible levels of description for physical fractals. In this context a fra ctal is a physical system which develops by interaction with the local envi ronment. We also mention that fractal space can be studied within the frame of a conformal relativity: As regards the dimensionality of spacetime, we start from the premiss that we cannot mix the four dimensions of spacetime with (possibly infinite) ext ra dimensions of quantum geometrodynamics. In the evolution of the universe these two types of dimensions originated at different stages of developmen t and different scales of distances. The situation is related to the fact t hat we cannot mix quantum or relativistic physics with classical physics. I t appears that it is superfluous to look for a demonstration of the asserti on that our spacetime is four dimensional. In fact, 'spacetime is 4D' is no t a theorem but a fundamental axiom or principle confirmed by empirical fac ts like any other axiom we apply in physics. Following El Naschie's conjecture that gravity is a phenomenon caused by ti me flowing at varying speeds ('multiple-time scale of time'), we elaborate the concept of time-gravitation induction: 'a (nonintegrable) variation of time flow generates gravitation and a variation of gravitation generates va riable time flow', a (dual) phenomenon similar to electro-magnetic inductio n duality. This idea is generalised and it is shown that focal (nonintegrab le, anholonomic and anisotropic) scale transformations lead to new local (f ractal) fields which may explain the generation of physical fields by the f ractal structure of spacetime. The existence of empty waves (i.e., a sort o f de Broglie waves separated from the associated particle) is also a conseq uence of an intrinsically fractalized spacetime. Special attention is payed to the interpretation of a double-slit experimen t and Feynman's path integral. It is shown that geometric excitons can act as diffusion micro-polarizers and may lead, consequently, to the fractaliza tion land thus to a nondifferentiability) of a trajectory. A fundamental co nclusion of the present paper is that nature (space, time, matter and field s) does not fractalize, it is intrinsically fractal. In fact this is the le itmotiv and raisond'etre of our work. (C) 2000 Elsevier Science Ltd. All ri ghts reserved.