The theory of scale relativity: Non-differentiable geometry and fractal space-time

The aim of the theory of scale relativity is to derive the physical behavior of a non-differentiable and fractal space-time and of its geodesics (with which particles are identified), under the constraint of the principle of the relativity of scales. We mainly study in this contribution the effects induced by internal fractal structures on the motion in standard space. We find that the main consequence is the transformation of classical mechanics in a quantum mechanics. The various mathematical quantum tools (Complex wave functions, spinors, bi-spinors) are built as manifestations of the non-differentiable geometry. Then the Schrodinger, Klein-Gordon and Dirac equations are successively derived as integrals of the geodesics equation, for more and more profound levels of description. Finally we tentatively suggest a new development of the theory, in which quantum laws would hold also in the scale-space: in such an approach, one naturally defines a new conservative quantity, named 'complexergy', which measures the complexity of a system as regards its internal hierarchy of organization. We also give some examples of applications of these proposals in various sciences, and of their experimental and observational tests.