  The development of a mechanics of non-differentiable paths suggested by Scale
Relativity results in a foundation of Quantum Mechanics including
Schr\"odinger's equation and all the other axioms under the assumption the path
non-differentiability can be described as a Wiener process at the
resolution-scale of observation. This naturally brings under question the
possibility that the statistics of the dynamics of macroscopic systems
fulfilling this hypothesis could fall under a quantum-like description with the
Planck constant replaced with some other constant, possibly system specific,
and corresponding to a diffusion coefficient. The observation of such a
quantum-like dynamics would establish if the Scale Relativistic principle is
implemented in macroscopic complex or chaotic systems. This would have major
implications for the study of structure formation dynamics in various research
fields. In this paper, I investigate the possibility for the detection of such
an effect in the Brownian motion of a micro-sphere in an optical trap. I find
that, if it exists, the observation of the transition to a quantum-like regime
is within reach of modern experiments.
