Wave–particle duality through an extended model of the scale relativity theory

Abstract

Considering that the chaotic effect of associated wave packet on the particle itself results in movements on the fractal (continuous and non-differentiable) curves of fractal dimension DF, wave–particle duality through an extension of the scale relativity theory is given. It results through an equation of motion for the complex speed field, that in a fractal fluid, the convection, dissipation and dispersion are reciprocally compensating at any scale (differentiable or non-differentiable). From here, for an irrotational movement, a generalized Schrödinger equation is obtained. The absence of dispersion implies a generalized Navier–Stokes type equation, whereas, for the irrotational movement and the fractal dimension, DF = 2, the usual Schrödinger equation results. The absence of dissipation implies a generalized Korteweg–de Vries type equation. In such conjecture, at the differentiable scale, the duality is achieved through the flowing regimes of the fractal fluid, i.e. the wave character by means of the non-quasi-autonomous flowing regime and the particle character by means of the quasi-autonomous flowing regime. These flowing regimes are separated by ‘0.7 structure’. At the non-differentiable scale, a fractal potential acts as an energy accumulator and controls through the coherence the duality. The correspondence between the differentiable and non-differentiable scales implies a Cantor space-time. Moreover, the wave–particle duality implies at any scale a fractal.