Grounding Bohmian Mechanics in Weak Values and Bayesianism

TL;DR

This paper links Bohmian mechanics to weak values and Bayesian probability, showing how to experimentally determine particle trajectories and derive quantum probabilities from an operational perspective.

Contribution

It introduces a method to uniquely determine Bohmian velocities via weak measurements and connects quantum probabilities to Bayesian principles.

Findings

The unique Bohmian velocity can be experimentally obtained as a weak value.

Bohmian paths can be reconstructed from large ensembles.

Quantum probabilities can be derived from Bayesian reasoning.

Abstract

Bohmian mechanics (BM) is a popular interpretation of quantum mechanics in which particles have real positions. The velocity of a point x in configuration space is defined as the standard probability current j(x) divided by the probability density P(x). However, this ``standard'' j is in fact only one of infinitely many that transform correctly and satisfy \dot P + \del . j=0. In this article I show that there is a unique j that can be determined experimentally as a weak value using techniques that would make sense to a classical physicist. Moreover, this operationally defined j equals the standard j, so, assuming \dot x = j/P, the possible Bohmian paths can also be determined experimentally from a large enough ensemble. Furthermore, this approach to deriving BM singles out x as the hidden variable, because (for example) the operationally defined momentum current is in general incompatible with the evolution of the momentum distribution. Finally I discuss how, in this setting, the usual quantum probabilities can be derived from a Bayesian standpoint, via the principle of indifference.