Emergence of complex and spinor wave functions in Scale Relativity. II. Lorentz invariance and bi-spinors

TL;DR

This paper demonstrates how complex and spinor wave functions naturally emerge from the non-differentiable, fractal space-time framework of Scale Relativity, providing a foundational derivation of relativistic quantum equations like Klein-Gordon and Dirac.

Contribution

It offers a first-principles derivation of Lorentz invariance and relativistic wave equations within the Scale Relativity theory, emphasizing the natural emergence of bi-spinors and Lorentz-invariant expectation values.

Findings

Bi-spinors naturally emerge from the theory.

Lorentz invariance of expectation values is derived within the framework.

Revisits derivations of Klein-Gordon and Dirac equations from first principles.

Abstract

Owing to the non-differentiable nature of the theory of Scale Relativity, the emergence of complex wave functions, then of spinors and bi-spinors occurs naturally in its framework. The wave function is here a manifestation of the velocity field of geodesics of a continuous and non-differentiable (therefore fractal) space-time. In a first paper (Paper I), we have presented the general argument which leads to this result using an elaborate and more detailed derivation than previously displayed. We have therefore been able to show how the complex wave function emerges naturally from the doubling of the velocity field and to revisit the derivation of the non relativistic Schr\"odinger equation of motion. In the present paper (Paper II) we deal with relativistic motion and detail the natural emergence of the bi-spinors from such first principles of the theory. Moreover, while Lorentz invariance has been up to now inferred from mathematical results obtained in stochastic mechanics, we display here a new and detailed derivation of the way one can obtain a Lorentz invariant expression for the expectation value of the product of two independent fractal fluctuation fields in the sole framework of the theory of Scale Relativity. These new results allow us to enhance the robustness of our derivation of the two main equations of motion of relativistic quantum mechanics (the Klein-Gordon and Dirac equations) which we revisit here at length.