Variance Decomposition in Bohmian Mechanics with Weak Actual Value Field and Quantum Potential

Abstract: We introduce a trajectory-based decomposition of quantum variances within Bohmian mechanics. By extending the weak actual value to a field on configuration space, we prove, under strong regularity conditions for stationary bound states, that the standard quantum variance splits into two non-negative terms: the ensemble variance of weak actual value and a quantum termarising from phase-amplitude coupling. For momentum, linking variance-level fluctuations to the average quantum potential. The decomposition fails to provide a physical interpretation for spin, reinforcing the Bohmian tenet that only position is fundamental. The work provides a formal tool for analyzing quantum fluctuations and clarifies the interpretative limits of such a trajectory-based approach.