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 $ψ=R\exp(iθ/\hbar)$, 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 $S=\partial_μA^μ$ is sourced by the extra-current $I=\partial_μj^μ$, 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.

• The paper shows that the Bohm quantum potential and the Aharonov–Bohm scalar mode are structurally linked through the macroscopic wavefunction’s amplitude profile.
• It utilizes the Madelung decomposition to elucidate how spatial inhomogeneity mediates quantum pressure and scalar electromagnetic responses.
• The analysis reveals that the relationship, sensitive to boundary conditions and normalization, provides a diagnostic framework bridging quantum hydrodynamics and electromagnetic theory.

Structural Connection Between the Bohm Quantum Potential and the Scalar Aharonov–Bohm Mode in Bosonic Schrödinger Models

Overview

This work investigates the formal and physical connection between the Bohm quantum potential ($Q_B$) and the scalar mode ($S = \partial_{\mu}A^{\mu}$) in the extended Aharonov–Bohm (AB) electrodynamics, within the framework of a non-relativistic bosonic Schrödinger model. The primary finding is that both $Q_B$ and the source of $S$ are structurally linked via their dependence on the spatial amplitude profile $R$ of the macroscopic wavefunction, as represented in the Madelung decomposition. The paper elucidates the mediated functional relationship between $S$ and $Q_B$, formulates the conditions for this dependence, and clarifies the operational and information-theoretic meaning of $Q_B$ in this context.

Theoretical Framework

Bohm Quantum Potential in the Madelung Decomposition

The quantum wavefunction $\psi = R e^{i\theta/\hbar}$ admits a hydrodynamic representation; the amplitude $R$ and phase $S = \partial_{\mu}A^{\mu}$0 yield current density and local energy balance equations. The quantum potential,

$S = \partial_{\mu}A^{\mu}$1

emerges as a state-dependent term representing the curvature of $S = \partial_{\mu}A^{\mu}$2 in the Hamilton–Jacobi form of the Schrödinger equation. Unlike classical potentials, $S = \partial_{\mu}A^{\mu}$3 is nonlocal and encodes kinetic energy contributions due to quantum inhomogeneity.

Scalar Mode in Extended Aharonov–Bohm Electrodynamics

The AB extension promotes $S = \partial_{\mu}A^{\mu}$4 to a physical, dynamical scalar. Instead of coupling only to locally conserved currents, its dynamics are sourced by an "extra-current" $S = \partial_{\mu}A^{\mu}$5, which, in the effective bosonic model, is directly related to the density-weighted divergence $S = \partial_{\mu}A^{\mu}$6. This modification was originally motivated for bosonic effective theories where scalar charged condensates (or quasiparticle ensembles) generate nontrivial electromagnetic back-reaction [(2605.10986), Minotti & Modanese].

Mediated Structural Linkage

The principal result is a formal, boundary-condition-dependent functional connection between $S = \partial_{\mu}A^{\mu}$7 and $S = \partial_{\mu}A^{\mu}$8:

• $S = \partial_{\mu}A^{\mu}$9 is uniquely determined by the amplitude curvature $Q_B$0, but is invariant under rescalings of $Q_B$1.
• The source term for $Q_B$2 depends quadratically on $Q_B$3 as $Q_B$4, which is sensitive to both inhomogeneity and absolute normalization.

The functional dependence is expressed via:

$Q_B$5

which allows the rewrite,

$Q_B$6

The relation is not local or invertible without boundary and normalization data, and it breaks down in regions where $Q_B$7 vanishes (e.g., at condensate nodes). While $Q_B$8 does not act as an explicit source for $Q_B$9, both are interpretable as functionals of the condensate amplitude $S$0.

Physical and Information-Theoretical Implications

The Bohm potential's physical status is clarified:

• $S$1 is not an external field but a local, state-dependent marker of the amplitude's texture.
• Integration over space reveals that

  $S$2

which represents the amplitude-gradient (texture) energy.

• For normalized densities, this is proportional to the Fisher information,

  $S$3

indicating that $S$4 quantifies the information-theoretic "sharpness" of the condensate profile.

Thus, $S$5 serves as a measurable diagnostic for quantum pressure, rigidity, and inhomogeneity—the same amplitude inhomogeneity that sources the scalar electromagnetic mode $S$6.

Numerical and Analytical Reductions

The paper provides explicit reductions in both one-dimensional and ansatz-based settings:

• In $S$7D, the relations simplify to Riccati-type ODEs for $S$8 given $S$9, with $R$0 computed as an explicit function of amplitude geometry and vector potential.
• For amplitude profiles satisfying geometric ansätze (e.g., $R$1), the mediation between $R$2 and $R$3 is analytically tractable.

Limitations and Scope

The formalism is subject to important caveats:

• The relationship between $R$4 and $R$5 is not a direct causation, but a mediated linkage through the amplitude profile $R$6.
• $R$7 is invariant under $R$8, whereas the source of $R$9 is sensitive to the normalization and global scale of $S$0.
• Specification of physical boundary data and attention to nodes in the amplitude are essential for unique construction.
• The connection is structural and diagnostic, not mechanistic.

Implications for Quantum Electrodynamic Theory

The identification of a structural bridge between Bohmian quantum hydrodynamics and the extended AB scalar mode invites new investigations into how quantum inhomogeneity—manifested as quantum pressure, gradient energy, and Fisher information—can have electromagnetic ramifications in effective matter models. This has potential implications for:

• Diagnostics of condensate states in which electromagnetic field interactions are noncanonical or not strictly gauge-invariant (e.g., superconductors, superfluids).
• Geometric/informational interpretations of quantum states in field-theoretic frameworks.
• Development of electromagnetic response theories sensitive to quantum statistical and geometric properties of order-parameter fields.

Conclusion

The paper establishes that the scalar mode in extended Aharonov–Bohm electrodynamics and the Bohm quantum potential are structurally connected by their mutual dependence on the order-parameter amplitude $S$1 in effective bosonic Schrödinger systems. While $S$2 and $S$3 do not engage in a direct causal relationship, their linkage through the spatial inhomogeneity of $S$4 provides a unified diagnostic framework. This connection enhances the interpretation of quantum texture energy, bridges hydrodynamic and electromagnetic theories, and opens new avenues for both theoretical investigation and experimental characterization of inhomogeneous quantum condensates (2605.10986).