Scale Relativity: First Steps toward a Field Theory

After having briefly recalled the principles and methods on which the theory of scale relativity is founded, we apply our scale-covariant methods to the problem of recovering relativistic quantum mechanics. The withdrawal of the hypothesis of differentiability of space-time leads to the demonstration of the divergence of spacetime coordinates in terms of resolutions, then to the definition of scale-covariant derivatives that transform the free particle classical equation into the Klein-Gordon equation, including electromagnetic potentials. Our suggestion of a generalization of the dilation laws to a Lorentzian form at high energies and its consequence for the generation of elementary particle masses is reminded. We conclude by new theoretical predictions, concerning in particular the W/Z boson mass ratio, which we expect to be √(10/13).