A Structural Link Between the Bohm Quantum Potential and the Scalar Mode of Aharonov-Bohm Electrodynamics in a Bosonic Schrödinger 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 $\ddot{I}ˆ=R\exp (i\hat{I}¸/\hslash )$, the Bohm quantum potential $Q_B=-\frac{{\hslash }^2}{2m}\frac{{\nabla }^{2}R}{R}$ is determined by the relative curvature ${\nabla }^{2}R/R$ of the amplitude profile $R$. In the same bosonic model, the scalar electromagnetic mode $S={\partial }_{\hat{I}}¼A^{\hat{I}}¼$ is sourced by the extra-current $I={\partial }_{\hat{I}}¼j^{\hat{I}}¼$, 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.

11 pages, no figures