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The development of a mechanics of non-differentiable paths suggested by ScaleRelativity results in a foundation of Quantum Mechanics includingSchr\"odinger's equation and all the other axioms under the assumption the pathnon-differentiability can be described as a Wiener process at theresolution-scale of observation. This naturally brings under question thepossibility that the statistics of the dynamics of macroscopic systemsfulfilling this hypothesis could fall under a quantum-like description with thePlanck constant replaced with some other constant, possibly system specific,and corresponding to a diffusion coefficient. The observation of such aquantum-like dynamics would establish if the Scale Relativistic principle isimplemented in macroscopic complex or chaotic systems. This would have majorimplications for the study of structure formation dynamics in various researchfields. In this paper, I investigate the possibility for the detection of suchan effect in the Brownian motion of a micro-sphere in an optical trap. I findthat, if it exists, the observation of the transition to a quantum-like regimeis within reach of modern experiments.

The possibility of Scale Relativistic signatures in the Brownian motion of micro-spheres in optical traps