Actual and weak actual values in Bohmian mechanics

TL;DR

This paper explores the concept of weak actual values in Bohmian mechanics, linking them to weak measurement theory, and clarifies their physical interpretation through theoretical analysis and application to experiments.

Contribution

It introduces weak actual values in Bohmian mechanics, establishes their relation to weak values, and clarifies their physical significance in experimental contexts.

Findings

Ensemble average of weak actual values equals quantum expectation value

Derived exact time evolution equation for weak actual values

Clarified the roles of real and imaginary parts of weak values in experiments

Abstract

We systematically analyze Holland's local expectation values within Bohmian mechanics, referring to them as weak actual values to emphasize their connection with weak measurement theory. These quantities are derived constructs that characterize local features associated with observables along Bohmian trajectories. We prove that their ensemble average equals the quantum expectation value and derive their exact time evolution equation. We formally establish their precise relation to the real part of the weak value in quantum measurement theory under position postselection, generalizing earlier insights. Applying this framework to a recent waveguide experiment clarifies the distinct physical roles of the real and imaginary parts of weak values, resolving an apparent challenge to Bohmian mechanics.