A Structural Link Between the Bohm Quantum Potential and the Scalar Mode of Aharonov-Bohm Electrodynamics in a Bosonic Schr\"odinger Model

Abstract

We discuss a formal and physical connection between the Bohm quantum potential and the scalar mode of the Aharonov-Bohm extension of electrodynamics. The analysis is motivated by the effective non-relativistic bosonic model recently proposed by Minotti and Modanese, in which the electromagnetic field is coupled to a conserved current while the field equations contain an additional source term. In the Madelung representation $\psi=R\exp(i\theta/\hbar)$, the Bohm quantum potential $ Q_B=-\frac{\hbar^2}{2m}\frac{\nabla^2 R}{R} $ is determined by the relative curvature $\nabla^2R/R$ of the amplitude profile $R$. In the same bosonic model, the scalar electromagnetic mode $S=\partial_\mu A^\mu$ is sourced by the extra-current $I=\partial_\mu j^\mu$, which contains the density-weighted electromagnetic combination $\nabla\cdot(R^2\mathbf A)$. Thus $Q_B$ does not act as a direct source of $S$; rather, the two quantities probe different differential aspects of the same amplitude profile: $Q_B$ is sensitive to the relative curvature of $R$, whereas the source of $S$ is sensitive to its density and gradient content through $R^2$ and $\nabla R$. We show that, once boundary and normalization data are fixed, this observation may be written as a mediated functional dependence of $S$ on $Q_B$ through $R$. We also clarify the physical status of $Q_B$: although it is state-dependent and should not be interpreted as an autonomous external potential, its density-weighted integral gives the amplitude-gradient energy, equivalently a Fisher-information contribution. This makes $Q_B$ a compact diagnostic of quantum pressure, rigidity, and inhomogeneity of a bosonic condensate. The resulting link with $S$ is therefore best understood as a structural relation between the order-parameter amplitude profile of the condensate and the scalar sector of the extended electromagnetic theory.