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

Abstract

The present paper explores the physical structure of two fundamental concepts, spacetime and fractals, both of which suffer from a lack of a satisfactory definition. We carefully consider the essential properties of these concepts and attempt to answer the questions: Have space and time an independent existence (with respect to matter and fields) and can we speak of a fractal 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 it is of interest to determine at how many different levels fractal spacetime 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 strong 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 still study their properties applying the Finsler theory of sprays which approximates physical fractals and may have the same general properties as ideal fractals (self-similarity, self-affinity, etc). This point of view is further supported by generalising the concepts of allometry and hierarchy of possible levels of description for physical fractals. In this context a fractal is a physical system which develops by interaction with the local environment. 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) extra dimensions of quantum geometrodynamics. In the evolution of the universe these two types of dimensions originated at different stages of development and different scales of distances. The situation is related to the fact that we cannot mix quantum or relativistic physics with classical physics. It appears that it is superfluous to look for a demonstration of the assertion that our spacetime is four dimensional. In fact, ` spacetime is 4D' is not a theorem but a fundamental axiom or principle confirmed by empirical facts like any other axiom we apply in physics. Following El Naschie's conjecture that gravity is a phenomenon caused by time 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 variable time flow', a (dual) phenomenon similar to electro-magnetic induction duality. This idea is generalised and it is shown that local (nonintegrable, anholonomic and anisotropic) scale transformations lead to new local (fractal) fields which may explain the generation of physical fields by the fractal structure of spacetime. The existence of empty waves (i.e., a sort of de Broglie waves separated from the associated particle) is also a consequence of an intrinsically fractalized spacetime. Special attention is payed to the interpretation of a double-slit experiment and Feynman's path integral. It is shown that geometric excitons can act as diffusion micro-polarizers and may lead, consequently, to the fractalization (and thus to a nondifferentiability) of a trajectory. A fundamental conclusion of the present paper is that nature (space, time, matter and fields) does not fractalize, it is intrinsically fractal. In fact this is the leitmotiv and raisond'être of our work.