Hall's exact variance decomposition in Bohmian Mechanics

Abstract

Halls exact variance decomposition [Phys. Rev. A 64, 052103 (2001)] splits the quantum variance of an observable into the ensemble variance of an optimal position based estimate and a residual nonclassical inaccuracy. We evaluate this decomposition in Bohmian mechanics. For momentum, the optimal estimate coincides with the Bohmian guidance field, and the inaccuracy is proportional to the ensemble average of the quantum potential. This gives a variance level identity separating momentum fluctuations into classical statistical dispersion and a quantum contribution from amplitude variations. The real and imaginary parts of the weak value map directly onto the two decomposition terms. By contrast, the inaccuracy vanishes for spin. This distinction is traced to the kinematic status of velocity in the primitive ontology, showing how the decomposition distinguishes observables dynamically coupled to local beables from merely contextual ones.