Scale relativity and fractal space-time: Applications to quantum physics, cosmology and chaotic systems

The theory of scale relativity is a new approach to the problem of the origin of fundamental scales and of scaling laws in physics, that consists of generalizing Einsteinʹs principle of relativity (up to now applied to motion laws) to scale transformations. Namely, we redefine space-time resolutions as characterizing the state of scale of the reference system and require that the equations of physics keep their form under resolution transformations (i.e. be scale covariant). We recall in the present review paper how the development of the theory is intrinsically linked to the concept of fractal space-time, and how it allows one to recover quantum mechanics as mechanics on such a non-differentiable space-time, in which the Schrödinger equation is demonstrated as a geodesies equation. We recall that the standard quantum behavior is obtained, however, as a manifestation of a “Galilean” version of the theory, while the application of the principle of relativity to linear scale laws leads to the construction of a theory of special scale relativity, in which there appears impassable, minimal and maximal scales, invariant under dilations. The theory is then applied to its preferential domains of applications, namely very small and very large length- and time-scales, i.e. high energy physics, cosmology and chaotic systems.