  Gauge field theory is developed in the framework of scale relativity. In this
theory, space-time is described as a non-differentiable continuum, which
implies it is fractal, i.e., explicitly dependent on internal scale variables.
Owing to the principle of relativity that has been extended to scales, these
scale variables can themselves become functions of the space-time coordinates.
Therefore, a coupling is expected between displacements in the fractal
space-time and the transformations of these scale variables. In previous works,
an Abelian gauge theory (electromagnetism) has been derived as a consequence of
this coupling for global dilations and/or contractions. We consider here more
general transformations of the scale variables by taking into account separate
dilations for each of them, which yield non-Abelian gauge theories. We identify
these transformations with the usual gauge transformations. The gauge fields
naturally appear as a new geometric contribution to the total variation of the
action involving these scale variables, while the gauge charges emerge as the
generators of the scale transformation group. A generalized action is
identified with the scale-relativistic invariant. The gauge charges are the
conservative quantities, conjugates of the scale variables through the action,
which find their origin in the symmetries of the ``scale-space''. We thus found
in a geometric way and recover the expression for the covariant derivative of
gauge theory. Adding the requirement that under the scale transformations the
fermion multiplets and the boson fields transform such that the derived
Lagrangian remains invariant, we obtain gauge theories as a consequence of
scale symmetries issued from a geometric space-time description.
