Volume 17: Pages 177-194, 2004
Physical Nature of the Bohmian Quantum Potential
We note that the Bohmian “quantum” potential, which, when added to the equations of classical mechanics, transforms these latter into the equations of quantum mechanics, is not addressed successfully in the literature so as to assign to it a fundamental physical source but is rather the invariable result of isolating it as a “suggestive” term from an arbitrary decomposition of the Schrödinger equation (SE). We will introduce a “hidden‐variable”mechanism for establishing the functional form of the Bohmian quantum potential and thus describe the single‐particle SE in causal terms for its “observable” particle. We only require that this Bohmian particle be understood as the ensemble average at (r, t) of the unobservable properties of many “hidden‐from‐experiment” identical particles there. We consider a restricted interaction between two classical ensembles, two distinct species of mass‐point particles, the very heavy macrons and the much lighter microns, as it occurs only between members of different species. We then imagine the behavior of an averaged macron as mimicking a quantum particle's motion when the former is bombarded by a myriad of microns that supposedly mimic the elements of the classical zero‐point vacuum. We refer the average kinetic energy of our macrons with respect to a Galilean inertial frame and express it as half the mass times the average value of its squared velocity according to statistical principles. We next replace this form by an equivalent form that adds to the squared average velocity a term that is none other than the variance of the macron velocity. We will initially express the individual macron velocities as a deterministic environmentally derived partition and a random encounter‐derived partition. We can then relate the variance of the random partition velocity to the variance of the micron velocity at an average encounter event (r, t) by accounting for the energy and momentum conservation laws. We just call upon the generating function of probability theory to calculate for the macron the average squared random partition velocity relative to the squared average random partition velocity, where, like the micron random velocity, the average macron random partition velocity is identified with a diffusion process. We can do this because the stochastic process involved is stationary, indicating that the Bohmian equations reflect equilibrium physics upon ensemble averaging. We must realize that the sum of the two partition velocities does not have this property of diffusion, however, except in the limiting case where there are no environmental forces, so that the SE is not a diffusion equation in real space. We thus recover the Bohmian quantum potential as proportional to the variance of the random partition, which, stated otherwise, is the fluctuation in the macron random kinetic energy at (r, t). We must note that, whereas Bohmian mechanics follows the causal motion of one definite particle, ours is a statistical theory of many typical particles and does not center on a basic one. We repeat, in closing, that our quantum potential, as a physical “cause,” is conceptually preferable to the ad hoc nonphysical entity that it currently manifests as the de Broglie “guiding” wave, a “mysterious” entity that acts directly on a quantum particle but is itself not affected.
Keywords: Bohmian mechanics, Brownian motion, causal quantum theory, continuous trajectories, distant universe effect, gravitons, guiding wave, hidden‐variable theories, indeterminacy, measurement, nonlocality property, ontology of physics, quantum potential, variance of velocity
Received: October 22, 2003; Published online: December 15, 2008