Wave-Particle Duality through a Hydrodynamic Model of the Fractal Space-Time Theory

Abstract

Considering that the microparticle movements take place on fractal curves, the wave–particle duality is studied in the fractal space-time theory (scale relativity theory). The Nottale model was extended by assuming arbitrary fractal dimension, DF, of the fractal curves and third-order terms in the equation of motion of a complex speed field. It results that, in a fractal fluid, the convection, dissipation, and dispersion are reciprocally compensating at any scale (differentiable or non-differentiable), whereas a generalized Schrodinger equation is obtained for an irrotational movement of the fractal fluid. The absence of the dispersion implies a generalized Navier–Stokes type equation and the usual Schrodinger equation results for the irrotational movement in DF = 2 of the fractal fluid. The absence of dissipation implies a generalized Korteweg–de Vries type equation. In such conjecture, the duality is analyzed through a hydrodynamic formulation. At the differentiable scale, the duality is achieved by the flowing regimes of the fractal fluid, while at the non-differentiable scale, a fractal potential controls, through the coherence, the duality.