When considering time and space scales such that the description of the trajectories of these bodies is in the shape of non-differentiable curves, one gets fractal curves of fractal dimension 2. Continuity and non-differentiability yield a fractal space and a symmetry breaking of the differential time element which gives a doubling of the velocity fields. An application of these principles on the motion equation of free particles leads to the occurrence of a supplementary TISE (time independent, Schr dinger-type equation). When using a WKBJ method to solve this equation, one gets some interesting results. In other words, scale transformation laws produce, on the motion equation of free particles under some peculiar assumptions, effects which are analogous to those of a "macroscopic quantum mechanics".
Navier-Stokes equations in scale relativity theory predict that the (vector) velocity field Fand the (scalar) density field p, on small distances (the same magnitude as the Compton length) behave like wave functions and are transported by the motion of a Newtonian fluid with velocity U. When considering vacuum from Casimir cavities (either two metallic slabs, parallel to each other, placed at a distance d apart or rectangular enclosures of sides d h d) a non-differentiable, Newtonian, 2D non-coherent quantum fluid whose entities assimilated to vortex-type objects become coherent (the coherence of the quantum fluid reduces to its ordering in vortex streets, being trigered by cvasiparticles) due to the constraints induced by the presence of walls. In other words, non-differentiability and coherence of the quantum fluid due to constraints, generate pressure. Casimir type forces are derived which are in good agreement with other theoretical results and experimental data.
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