On Nonlinear Quantum Mechanics, Brownian Motion, Weyl Geometry and Fisher Information

A new nonlinear Schrödinger equation is obtained explicitly from the (fractal) Brownian motion of a massive particle with a complex-valued diffusion constant. Real-valued energy plane-wave solutions and solitons exist in the free particle case. One remarkable feature of this nonlinear Schrödinger equation based on a ( fractal) Brownian motion model, over all the other nonlinear QMmodels, is that the quantum-mechanical energy functional coincides precisely with the field theory one. We finalize by showing why a complex momentum is essential to fully understand the physical implications of Weyl’s geometry in QM, along with the interplay between Bohm’s Quantum potential and Fisher Information which has been overlooked by several authors in the past. PACS numbers: 03.65, 05.40.J, 47.53, 04.20.G