Can the wave function in configuration space be replaced by single-particle wave functions in physical space?

Abstract

The ontology of Bohmian mechanics includes both the universal wave function (living in 3N-dimensional configuration space) and particles (living in ordinary 3-dimensional physical space). Proposals for understanding the physical significance of the wave function in this theory have included the idea of regarding it as a physicallyreal field in its 3N-dimensional space, as well as the idea of regarding it as a law of nature. Here we introduce and explore a third possibility in which the configuration space wave function is simply eliminated-replaced by a set of single-particle pilotwave fields living in ordinary physical space. Such a re-formulation of the Bohmian pilot-wave theory can exactly reproduce the statistical predictions of ordinary quantum theory. But this comes at the rather high ontological price of introducing an infinite network of interacting potential fields (living in 3-dimensional space) which influence the particles' motion through the pilot-wave fields. We thus introduce an alternative approach which aims at achieving empirical adequacy (like that enjoyed by GRW type theories) with a more modest ontological complexity, and provide some preliminary evidence for optimism regarding the (once popular but prematurely-abandoned) program of trying to replace the (philosophically puzzling) configuration space wave function with a (totally unproblematic) set of fields in ordinary physical space.

Bohmian Conditional Wave Functions (and the status of the quantum state)

“…So this is in some sense a "small entanglement approximation" (SEA). [10] I illustrate this here with the simple example of the two interacting 1-dimensional SHOs mentioned before. This system is nice because it can be solved exactly.…”

Applied Bohmian Mechanics

“…One possible approach could be to follow the ideas introduced in Refs. [228,229] where a full (infinite) set of equations for an exact description of the conditional wave functions is presented.…”

“…Such tool is the conditional wave function [22,35,66,229,417]. We consider an isolated quantum system.…”

“…The total wave function cannot be written as a product Ψ (r) = ψ a (r a )ψ b (r b ) if the two subsystems are entangled. However, within Bohmian mechanics, we can define the so called conditional wave function [22,35,66,229,417]:…”

“…(29) from the universal big (many particle) wave function in the huge R 3N configuration space. In order to discuss the practical utility of these conditional wave functions, we need to explore the possibility of defining them independently of the big wave function [35,228,229]. We might track the dynamical evolution of a quantum (sub)system exclusively in terms of these single-particle (conditional) wave functions, instead of (as we did in their presentation) first solving the many-body Schrödinger equation for Ψ and only examining the conditional wave functions ψ after.…”