Manipulating and preparing additional variables
Franck Laloë · 2012
Introducing the hydrodynamic equations (Madelung [416]) associated with the evolution of the wave function, and using them to guide the evolution of the additional variables (positions of particles), is a natural idea. In the dynamics of fluids, hydrodynamic equations can be obtained by taking averages of microscopic quantities over positions and velocities of point-like particles; for instance, the Navier-Stokes macroscopic equations can be derived from the Boltzmann transport equation by appropriate microscopic averages (Chapman-Enskog method); conversely, the hydrodynamic variables will influence the motion of individual particles. Moreover, there is some analogy between the guiding term and the force term in a Landau type kinetic equation, where each particle is subject to an average force proportional to the gradient of the density of the others. Nevertheless, here we are dealing with a single particle, so that the guiding term cannot be associated with interactions between particles. Moreover, we also know from the beginning that rather unusual properties must be contained in the guiding equations, at least if we wish to exactly reproduce the predictions of usual quantum mechanics: the Bell theorem states that the additional variables have to evolve non-locally in ordinary three-dimensional space (they evolve locally only in the configuration space of the system, exactly as for the state vector). In other words, in real space the additional variables must be able to influence each other at an arbitrary distance. Indeed, in the Bohmian equation of motion of the additional variables, the velocity of a particle contains an explicit dependence on its own position, as expected, but also a dependence on the positions of all the other particles (assuming that the particles are entangled).