Riccati equations as a scalerelativistic gateway to quantum mechanics
Abstract
Applying the resolutionscale relativity principle to develop a mechanics of nondifferentiable dynamical paths, we find that, in one dimension, stationary motion corresponds to an Ito process driven by the solutions of a Riccati equation. We verify that the corresponding FokkerPlanck equation is solved for a probability density corresponding to the squared modulus of the solution of the Schrodinger equation for the same problem. Inspired by the treatment of the onedimensional case, we identify a generalization to time dependent problems in any number of dimensions. The Ito process is then driven by a function which is identified as establishing the link between nondifferentiable dynamics and standard quantum mechanics. This is the basis for the scale relativistic interpretation of standard quantum mechanics and, in the case of applications to chaotic systems, it leads us to identify quantumlike states as characterizing the entire system rather than the motion of its individual constituents.
 Publication:

arXiv eprints
 Pub Date:
 March 2019
 arXiv:
 arXiv:1904.05739
 Bibcode:
 2019arXiv190405739N
 Keywords:

 Physics  General Physics
 EPrint:
 16 pages, no figure