Scale relativity and gauge invariance∗

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. As a consequence, spacetime becomes fractal, i.e., explicitly resolution-dependent. The requirement that this geometry satisfies the principle of scale relativity leads to introduce scale laws having a Galilean form (constant fractal dimension), then a logLorentzian form. In this framework, the Planck length-time scale becomes a minimal impassable scale, invariant under dilations. Then we attempt to construct a generalized scale relativity which includes scale-motion coupling. In this last framework, one can reinterpret gauge invariance as scale invariance on the internal resolutions. This approach allows one to set new constraints in the standard model, concerning in particular the Higgs boson mass, which we find to be √ 2mW = 113.73 ± 0.06 GeV in a large class of models.