Non-Abelian gauge field theory in scale relativity

TL;DR

This paper develops a non-Abelian gauge field theory within the framework of scale relativity, where space-time's fractal nature and scale transformations lead to gauge fields and charges as geometric and symmetry-derived entities.

Contribution

It extends previous work by deriving non-Abelian gauge theories from scale transformations in a fractal space-time framework, linking gauge fields to geometric scale symmetries.

Findings

Derivation of non-Abelian gauge fields from scale transformations.

Identification of gauge charges as generators of scale symmetry.

Recovery of covariant derivatives and gauge invariance from geometric principles.

Abstract

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.