Generalized guidance equation for peaked quantum solitons and effective gravity

Bouncing oil droplets have been shown to follow de Broglie-Bohm like trajectories and at the same time they exhibit attractive and repulsive pseudo-gravitation. We propose a model aimed at rendering account of these phenomenological observations. It inspires, in a more speculative approach, a toy model for quantum gravity. In studies concerning quantum wave mechanics, de Broglie-Bohm (dB-B) trajectories [10, 3, 17] remained during many years a rather confidential and academic topic, but they regained interest since they were realized in the lab. with artificial macroscopic systems, the bouncing oil droplets or walkers [4, 9], which were shown experimentally to follow dB-B like quantum trajectories. For instance, when the walker passes through one slit of a two-slit device, it undergoes the influence of its “pilot-wave” passing through the other slit, in such a way that, after averaging over many dB-B like trajectories, the interference pattern typical of a double-slit experiment is restored, despite of the fact that each walker passes through only one slit. Pseudogravitational interaction has also been reported between two droplets. In [9] for instance we can read: ...We find that, depending on the value of d, (d represents here the impact parameter of the collision)the interaction is either repulsive or attractive. When repulsive, the drops follow two approximately hyperbolic trajectories. When attractive, there is usually a mutual capture of the two walkers into an orbital motion similar to that of twin stars ... Our study was motivated by the double solution program [2, 10, 11] of Louis de Broglie, according to which the quantum wave function, solution of the linear Schrödinger equation does not contain all elements of reality concerning the quantum system. In de Broglie’s view, particles would consist of a peaked concentration of energy that would coexist with the linear wave function, and the latter would guide the former according to the de Broglie-Bohm guidance equation. Our work consists of an attempt to realize de Broglie’s program by adding another ingredient to it, which is that we suppose the existence of a non-linear self focusing potential (2) of gravitational nature which prohibits the spreading of the peaked soliton (solitary wave) representing the particle. Therefore we assume that the system evolves in time according to the non-linear equation (1). i~ ∂Ψ(t,x) ∂t = −~ ∆Ψ(t,x) 2m + V (t,x)Ψ(t,x) + V (Ψ)Ψ(t,x), (1) 1Aix Marseille Université, CNRS, Centrale Marseille, Institut Fresnel (UMR 7249),13013 Marseille, France.email: [email protected]