Scale Relativity
Scale Relativity
Prof. Dr. Saeed Naif Turki - Department of physics
eps.saeedn.turkisntr2006@uoanbar.edu.iq
The beginning of the last century has witnessed the advent of two important events in physics, namely; the formulation of the theory of relativity by Einstein and quantum mechanics by Schrödinger, Heisenberg, Dirac and others. While the theory of relativity (special and general) is well-founded on physical (geometrical) basis, quantum mechanics is considered as an axiomatic theory based on purely mathematical rules. It was first introduced as a non-relativistic theory. There is still a difficulty in understanding the connection between mathematical tools and the physical interpretation in quantum mechanics. Also, the appearance of the constant ? (Dirac constant) in the Schrödinger equation is considered as one of the mysteries of quantum mechanics. Another mystery is the wave nature of the solution to the Schrödinger and other relativistic quantum mechanical equations. This wave nature is axiomatically connected with the non-classical probabilistic behavior of quantum systems in analogy with the electromagnetic wave of classical electromagnetism and its connection with the probabilistic nature of photons in light fields with some differences.
Attempts to introduce Einstein’s principle of special relativity into quantum mechanics have resulted in relativistic quantum mechanical wave equations for relativistic particles such as the Klien-Gordon equation and the Dirac equation. More serious attempts to reconcile the principle of relativity with quantum mechanics have faced grave difficulties even though some real progress has been made in this direction. The reason for these difficulties is usually traced back to the different nature of the two fields stated at the beginning of this chapter, namely; the geometrical nature of relativity theory and axiomatic (non-geometrical) nature of quantum mechanics. Attempts to geometrize quantum mechanics to ease its twining with relativity theory in a single more general theory were also made based on more than one direction . A more serious attempt in this direction can be traced back to Feynman who studied the geometrical structure of quantum paths and showed that the trajectory of a quantum particle is continuous and non-differentiable. At that time the connection between non-differentiability and the concept of fractals and the modern field of fractal geometry was not well established. Later, Abbot and Wise reconsidered the problem of the geometrical structure of quantum paths in terms of the concept of fractals and they demonstrated that the trajectory of a quantum particle varies with the resolution. Hence, they showed that the fractal dimension D of this trajectory is 2. Going further in this direction, Ord considered fractal space-time as a geometric analogue of relativistic quantum mechanics. He proposed two field equations for the description of what he called fractalons based on a random walk in space-time trajectories and subsequently related these equations to the free particle Klien-Gordon and Dirac equations. Building on these geometrical concepts, and taking the fractal space-time concept more seriously into consideration, Nottale introduced his theory of scale relativity (ScR) to reformulate quantum mechanics from first principles. The theory of ScR extends Einstein's principle of relativity of motion to scale transformations of resolutions. In other words, it is based on giving up the axiom of differentiability of the space-time continuum. The new framework as formulated by Nottale generalizes the standard theory of relativity and includes it as a special case . Three consequences arise from this withdrawal of differentiability of space - time, namely;
(i)- The geometry of space-time must be fractal, i.e., explicitly resolution-dependent. This leads to non- classical behavior as a consequence.
(ii)- The geodesics of this non-differentiable space-time are themselves fractal and infinite in number.
(iii)- Time reversibility is broken at the infinitesimal level. This is again a behavior which has no analogue in classical systems. According to Nottale, the ScR approach is expected to apply not only at small scales (quantum domain), but also at very large space-time scales (cosmological domain) although with different interpretation. Nottale also shows the applicability of his ScR theory to systems in the middle, namely; complex systems Therefore; there are three domains for the application of this theory which are microphysics, cosmology and complex systems.