[Paper Review] Gauge field theory in scale relativity

Laurent Nottale, Marie-Noëlle Célérier+1 more|ArXiv.org|Jul 10, 2003

Advanced Mathematical Theories and Applications5 references3 citations

TL;DR

This paper derives gauge field theories from the geometric principles of scale relativity, where gauge transformations emerge from scale symmetries in a fractal spacetime. By treating scale variables as dynamical functions of spacetime coordinates, the authors show that gauge fields and charges arise naturally as geometric contributions to the action, recovering the covariant derivative and gauge invariance without postulating them a priori.

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.

Motivation & Objective

• To provide a physical foundation for gauge field theory ingredients—gauge fields, charges, and transformations—within the scale relativity framework.
• To extend previous work on Abelian U(1) gauge theory to non-Abelian gauge groups, such as SU(2) and SU(5).
• To show that gauge invariance and the structure of the covariant derivative arise from geometric variations in scale variables, not postulated axioms.
• To interpret the internal space of gauge theory as a non-absolute scale-space, analogous to velocity in relativistic mechanics.
• To establish a geometric origin for fermion and boson field transformations that preserve the Lagrangian under scale symmetries.

Proposed method

• Treat scale variables ηα(x,y,z,t) as dynamical functions of spacetime, generalizing the Compton wavelength ratio ρ = λ/ε.
• Apply the principle of relativity of scales, so that scale transformations are linked to spacetime displacements.
• Define the generalized action as a scale-relativistic invariant, including both spacetime and scale variables.
• Derive the covariant derivative by computing the total variation of scale variables under combined spacetime and scale transformations.
• Identify gauge charges as conserved quantities arising from Noether-like symmetries of the scale-space group.
• Apply the formalism to non-Abelian cases, including SU(2) for weak interactions and SU(5) for grand unification, showing consistency with known group structures.

Experimental results

Research questions

• RQ1 How can gauge fields be derived from geometric principles in a fractal spacetime rather than postulated?
• RQ2 What is the physical meaning of the internal space in gauge theory within the scale relativity framework?
• RQ3 How do gauge charges emerge from symmetries of the scale-space rather than being introduced ad hoc?
• RQ4 Can the standard form of the covariant derivative in non-Abelian gauge theory be derived from scale-relativistic geometry?
• RQ5 What is the role of scale transformations in unifying the description of fermions, bosons, and gauge invariance?

Key findings

• Gauge fields emerge as geometric contributions to the total variation of scale variables under spacetime and scale transformations.
• The covariant derivative of non-Abelian gauge theory is derived from first principles in scale relativity, without postulating it.
• Fermion multiplets and gauge bosons transform in a way that preserves the Lagrangian, ensuring gauge invariance through geometric consistency.
• The theory naturally accommodates SU(5) grand unification, with 24 massless degrees of freedom before symmetry breaking.
• The proton lifetime prediction in this framework is adjusted by a factor of 10^16 due to mG = mPlanck, bringing it into agreement with experimental limits.
• Gauge charges are identified as generators of scale transformation groups acting on the generalized scale-relativistic action, providing a geometric origin for conservation laws.