Scale relativity and fractal space-time: theory and applications

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.

Insights into Scale Relativity and Fractal Space-Time

The paper on the theory of scale relativity and fractal space-time by L. Nottale presents a comprehensive framework that extends the principles of relativity to scales, thereby offering a novel geometric foundation for understanding physical laws, including quantum mechanics and gauge field theories. This approach eschews the differentiability hypothesis of space-time, instead proposing a fractal and non-differentiable continuous geometry.

Key Contributions:

1. Fractal Geometry and Quantum Mechanics: The study postulates that the principles of scale relativity, when applied to the physical universe, necessitate a fractal geometry. This geometry is not merely a mathematical abstraction but a tangible aspect of physical reality, where scale-dependent phenomena manifest as fractal laws. The fractal nature of space-time underlies a generalized description of quantum mechanics, moving away from an axiomatic foundation towards one rooted in the relativity of scales. This generalized framework can derive fundamental equations like Schrödinger's and Dirac's equations from first principles.
2. Scale Differentiability and Covariant Derivatives: A continuous but non-differentiable structure implies that classical notions of smooth gradients and differentiabilities are lost, necessitating a redefinition of derivatives. The paper describes how to handle such a fractal structure by employing covariant operators that respect the non-differentiable nature of space-time. These operators are foundational in formulating scale dynamics that are consistent with the concept of relativity, extending the concepts of geodesics from the differentiable manifold of general relativity to a fractal space-time.
3. Applications Across Disciplines: The theory is applied not only within the domain of quantum mechanics but extends to astrophysics, biology, and earth sciences, among others. Notable applications include:
   • Predictive models for the QCD coupling and cosmological constants.
   • Explaining patterns in planetary system formations and predicting extrasolar planets' distributions.
   • Modelling biological evolution and societal development through log-periodic laws indicating punctuated equilibria.
   • Proposing criticality in geological phenomena such as earthquake rate deceleration and arctic ice melt predictions.
4. Implications for Interdisciplinary Research: The proposed scale relativity framework continues to garner interest for its potential to unify concepts across diverse scientific disciplines, offering a coherent approach to complexity theory and emergent phenomena. It lays groundwork for potentially novel insights into processes that appear to be scale-invariant or demonstrate fractal-like behaviors within large and small-scale structures across the cosmos and biosphere.
5. Future Outlook: The theory promises to pave the way for new research directions, particularly in exploring the implications of a non-differentiable manifold in understanding space-time at fundamental levels. It suggests that future work could significantly deepen understanding of the quantum-classical transition, influence theories of quantum gravity, and provide insights into the structured complexity observed in life sciences.

In conclusion, Nottale’s proposition of scale relativity as a guiding framework challenges conventional notions, offering intriguing possibilities that traverse traditional boundaries in science. This underscores the importance of scale and dimensionality in the conceptualization and evolution of natural laws, embracing the complexity of fractal geometries. As applications of this theory widen, it may fundamentally alter perspectives in theoretical physics and beyond, nurturing interdisciplinary methodologies that harness the inherent fractal nature of reality.