Gauge field theory in scale relativity

TL;DR

This paper derives gauge field theories from the principles of scale relativity, linking gauge fields to geometric scale transformations in a fractal space-time framework, and applies this to electroweak and grand unified theories.

Contribution

It provides a geometric and physical interpretation of gauge fields within scale relativity, deriving gauge invariance and charges from scale-space symmetries without axiomatic assumptions.

Findings

Gauge fields emerge as geometric contributions in scale space.

Gauge charges are linked to scale transformation generators.

The framework reproduces non-Abelian gauge covariant derivatives.

Abstract

The aim of the present article is to give physical meaning to the ingredients of standard gauge field theory in the framework of the scale relativity theory. Owing to the principle of the relativity of scales, the scale-space is not absolute. Therefore, the scale variables are functions of the space-time coordinates, so that we expect a coupling between the displacement in space-time and the dilation/contraction of the scale variables, which are identified with gauge transformations. The gauge fields naturally appear as a new geometric contribution to the total variation of the scale variables. The gauge charges emerge as the generators of the scale transformation group applied to a generalized action (now identified with the scale relativistic invariant) and are therefore the conservative quantities which find their origin in the symmetries of the scale-space. We recover the expression for the covariant derivative of non-Abelian gauge theory. Under the gauge transformations, the fermion multiplets and the boson field transform in such a way that the Lagrangian, which is here derived instead of being set as a founding axiom, remains invariant. We have therefore obtained gauge theories as a consequence of scale symmetries issued from a geometric fractal space-time description, which we apply to peculiar examples of the electroweak and grand unified theories.