A mathematically explicit extension of quantum hydrodynamics is developed in which a scalar coherence field ΦG modifies phase transport and is driven by the information-geometric structure of the probability density together with probability transport itself. The construction starts from the Madelung representation Ψ = √ρ eiS/ℏ, augments the phase gradient by the covariant shift ∇S ↦ ∇S − αℏ∇ΦG, and defines the dynamics through a variational (ρ, S) sector together with a dissipative Onsager-type relaxation law for ΦG. This yields a closed PDE system consisting of a continuity equation, a modified Hamilton–Jacobi equation, and a coherence-field equation that reduces in the overdamped limit to an elliptic closure. The source of ΦG is neither bare mass density nor bare probability density: it is a local Fisher-information density together with probability transport, so that ΦG measures coherence demand. The field equations are written explicitly; the coupling sign is fixed by an energy-stability argument, the transport-forcing sign by a dissipation argument, and the induced coherence-pressure term is computed exactly. A structural no-signalling condition is obtained by restricting the nonlinearity to Fisher-marginal source functionals that are blind to the entanglement Fisher excess under marginalisation. A coarse-grained Yukawa sector is shown to admit a conditional Newtonian limit under explicitly stated screening and additivity assumptions; this remains a conjectural macroscopic regime rather than a completed derivation of gravity. The paper closes with a falsifiable numerical programme—ablation studies, dispersion diagnostics, Fisher-information monitoring, and an explicit pilot solver scaffold—together with a claim-status ledger, a testability map, and a development roadmap that separate what is defined, what is derived, what is heuristic, and what remains unproven. Keywords: quantum hydrodynamics; Fisher–Rao geometry; Madelung dynamics; coherence field; no-signalling; information geometry.
The conserved probability current of the Schrodinger field admits a dimension-dependent gauge representation that provides a convenient setting for comparing hydrodynamic, topological and infrared structures. We use this representation to develop a unified treatment of several deformations of the Madelung momentum. BF couplings incorporate electromagnetic interactions, Berry and spin connections, projected non-abelian adiabatic data, and intrinsic phase holonomy within the same framework. In
(2+1) dimensions, the Chern-Simons deformation can be reduced to a nonlocal functional of the wavefunction. Its Coulomb-gauge form separates into a phase-sensitive geometric contribution and a density-dependent dynamical contribution; we show explicitly how this separation changes under a general gauge transformation and recover the topological braiding phase in the well-separated adiabatic regime. For systems with a spatial boundary, we derive the equal-time symplectic structure, the associated surface charges, and their boundary algebra. Finally, after a nonlinear interaction produces the Bogoliubov acoustic branch, the radiative memory can be expressed in the dual two-form variables. Its zero-average component is generated by a residual large gauge transformation, whereas the monopole component remains independent global data. This provides a gauge-theoretic description of the memory/asymptotic-symmetry sector of the acoustic infrared structure.
By linearizing the hydrodynamic equations associated with the Madelung transformation in the context of Bohmian dynamics, we derive an effective metric for the phase perturbation around a background state. We show that the metric previously obtained previously by the same authors arises naturally as a particular case of a more general structure, in which the complete functional variation of the quantum potential is retained and external classical potentials can be incorporated. This generalization preserves the dependence of the quantum potential on the density and its spatial derivatives, generating additional terms that introduce dispersive corrections and act as sources in the effective Klein-Gordon equation. To organize these contributions, we introduce differential operators that separate the local sector responsible for the effective geometry from the corrections associated with the complete functional structure of the quantum potential. We identify the conditions under which these additional terms become negligible, recovering homogeneous propagation and the effective geometry for the Gaussian wave packet of a free particle.
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=Rexp(iθ/ℏ), the Bohm quantum potential $ Q_B=-\frac{\hbar^2}{2m}\frac{\nabla^2 R}{R} $ is determined by the relative curvature
∇2R/R of the amplitude profile
R. In the same bosonic model, the scalar electromagnetic mode
S=∂μAμ is sourced by the extra-current
I=∂μjμ, which contains the density-weighted electromagnetic combination
∇⋅(R2A). Thus
QB does not act as a direct source of
S; rather, the two quantities probe different differential aspects of the same amplitude profile:
QB is sensitive to the relative curvature of
R, whereas the source of
S is sensitive to its density and gradient content through
R2 and
∇R. We show that, once boundary and normalization data are fixed, this observation may be written as a mediated functional dependence of
S on
QB through
R. We also clarify the physical status of
QB: 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
QB 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.
Madelung showed how the complex Schrödinger equation can be rewritten in terms of two real equations, one for the phase and one for the amplitude of the complex wave function, where both equations are not independent of each other, but coupled. Although these equations formally look like classical hydrodynamic equations, they contain all the information about the quantum system. Concerning the quantum mechanical uncertainties of position and momentum, however, this is not so obvious at first sight. We show how these uncertainties are related to the phase and amplitude of the wave function in position and momentum space and, particularly, that the contribution from the phase essentially depends on the position-momentum correlations. This will be illustrated explicitly using generalized coherent states as examples.
This paper develops a hydrodynamic derivation of nonrelativistic quantum mechanics within the Hydrodynamic Quantum Gravity (HQG) programme. Using the Madelung transformation, the Schrödinger equation is written in amplitude–phase form and shown to be exactly equivalent to (i) a continuity equation for a conserved density and (ii) an Euler-type momentum equation augmented by the “quantum potential” term. In this interpretation, the quantum potential is treated as an effective dispersive stress associated with structured vacuum dynamics, rather than as a purely formal mathematical artefact. The paper reviews how standard quantum features appear naturally in the hydrodynamic variables: interference and diffraction as phase-driven flow, stationary states as time-harmonic solutions, and quantisation as a consequence of single-valued phase (circulation) constraints. It also discusses how pilot-wave style dynamics can be expressed in this language, and surveys laboratory analogues (including walking-droplet experiments) as phenomenological motivation and intuition-building, without claiming identity between the analogue systems and the quantum vacuum. Interpretation note: “Superfluid/condensate” language is used here as an effective mathematical/analogue description of vacuum behaviour; the paper does not assert a material substrate or an operationally accessible preferred rest frame. This paper is part of a series. Companion preprints develop the gravitational and electromagnetic components of the HQG framework and discuss additional open problems and testable predictions. Discussion and feedback: https://github.com/Gptham123456/Hydrodynamic-Quantum-Gravity/discussions