Scale relativity, fractal space-time and morphogenesis of structures

The theory of scale relativity extends Einstein’s principle of relativity to scale transformations of resolutions. It is based on the giving up of the axiom of differentiability of the space-time continuum. Three consequences arise from this withdrawal. (i) The geometry of space-time becomes fractal, i.e., explicitly resolutiondependent : this allows one to describe a non-differentiable physics in terms of differential equations acting in the scale space. The requirement that these equations satisfy the principle of scale relativity leads to introduce scale laws having a Galilean form (constant fractal dimension), then a log-Lorentzian form. In this framework, the Planck length-time scale becomes a minimal impassable scale, invariant under dilations, and the cosmic length-scale (related to the cosmological constant) a maximal one. Recent measurements of the cosmological constant have confirmed the theoretically predicted value. Then we attempt to construct a generalized scale relativity which includes non-linear scale transformations and scale-motion coupling. In this last framework, one can reinterpret gauge invariance as scale invariance on the internal resolutions. This approach has allowed us to make theoretical predictions concerning coupling constants and elementary particle masses (electron, Higgs boson, vacuum energy of the Higgs field), which we update in the present contribution. These predictions are successfully checked using recently improved experimental values. (ii) The geodesics of a non-differentiable space-time are fractal and in infinite number: this leads one to use a fluid-like description and implies adding new terms in the differential equations of mean motion. (iii) Time reversibility is broken at the infinitesimal level: this can be described in terms of a two-valuedness of the velocity vector, for which we use a complex representation. Published in “Sciences of the Interface”, Proceedings of International Symposium in honor of O. Rössler, ZKM Karlruhe, 18-21 May 2000, Eds. H. Diebner, T. Druckrey and P. Weibel, (Genista, Tübingen), p. 38