Geometric interpretation of gauge fields in the theory of scale relativity∗

The question of the physical nature of gauge fields is revisited in the framework of the theory of scale relativity, first in the Abelian case (quantum electrodynamics), then in the non-Abelian case (SU(2) and more general fields). Space-time is described as a non-differentiable and fractal continuous manifold (i.e., it is explicitly dependent on the scale of resolution). The gauge fields can then be interpreted in a purely geometric way: they are the manifestations of the contraction and dilation of the scale variables induced by space-time displacements, at scales smaller than the Compton length of a particle. The theory also allows one to give a geometrical meaning to the charges: they are defined as the conservative quantities that originate from the new scale symmetries. Moreover, in the framework of special scale-relativity, where the Planck length becomes a minimal, impassable horizon-like scale, invariant under dilations, the quantization of charges is ensured because the possible scale ratios become limited. As a consequence we theoretically predict relations between coupling constants α and mass scales m of the form α ln(mI P /m) = 1, where mI P is the Planck mass. Some evidence for the existence of such relations in the experimental data is given. We conclude by recalling our theoretical prediction of a Higgs boson mass of √ 2 mW = 114 GeV.