Stochastic Quantum Hydrodynamic Model from the Dark Matter of Vacuum Fluctuations: The Langevin-Schr？dinger Equation and the Large-Scale Classical Limit

The work derives the quantum evolution in a fluctuating vacuum by introducing the related (dark) mass density noise into the Madelung quantum hydrodynamic model. The paper shows that the classical dynamics can spontaneously emerge on the cosmological scale allowing the realization of the classical system-environment super system. The work shows that the dark matter-induced noise is not spatially white and owns a well defined correlation function with the intrinsic vacuum physical length given by the De Broglie one. By departing from the quantum mechanics (the deterministic limit of the theory) in the case of microscopic systems whose dimension is much smaller than the De Broglie length, the model leads to the Langevin-Schrodinger equation whose friction coefficient is not constant. The derivation puts in evidence the range of application of the Langevin-Schrodinger equation and the approximations inherent to its foundation. Increasing the physical length of the system description, the classical physics can be achieved when the scale of the problem is much bigger both than the De Broglie length and the quantum potential range of interaction. The long-distance characteristics as well as the range of interaction of the non-local quantum potential are derived and analyzed in order to have a coarse-grained large-scale description. The process of measurement, in the large-scale classical limit, satisfies the minimum uncertainty conditions if interactions and information do not travel faster than the light speed, reconciling the quantum entanglement with the relativistic macroscopic locality.