Scale relativity and fractal space-time: theory and applications

TL;DR

This paper reviews the development of scale relativity theory based on fractal space-time, explores its geometric foundations, and discusses diverse scientific applications validated by observational data.

Contribution

It introduces a geometric framework for scale relativity that generalizes fractal laws and provides a new foundation for quantum mechanics and gauge theories.

Findings

Predictions of physical constants like QCD coupling and cosmological constant.

Applications to astrophysics, including planetary distances and star cycles.

Insights into Earth sciences, such as earthquake patterns and sea ice decline.

Abstract

In the first part of this contribution, we review the development of the theory of scale relativity and its geometric framework constructed in terms of a fractal and nondifferentiable continuous space-time. This theory leads (i) to a generalization of possible physically relevant fractal laws, written as partial differential equation acting in the space of scales, and (ii) to a new geometric foundation of quantum mechanics and gauge field theories and their possible generalisations. In the second part, we discuss some examples of application of the theory to various sciences, in particular in cases when the theoretical predictions have been validated by new or updated observational and experimental data. This includes predictions in physics and cosmology (value of the QCD coupling and of the cosmological constant), to astrophysics and gravitational structure formation (distances of extrasolar planets to their stars, of Kuiper belt objects, value of solar and solar-like star cycles), to sciences of life (log-periodic law for species punctuated evolution, human development and society evolution), to Earth sciences (log-periodic deceleration of the rate of California earthquakes and of Sichuan earthquake replicas, critical law for the arctic sea ice extent) and tentative applications to system biology.