Amplitude-dependent quantum hydrodynamics from a $\coth$-Madelung ansatz

This paper proposes a nonlinear extension of the Madelung transformation using a hyperbolic phase-amplitude coupling to derive amplitude-dependent quantum hydrodynamics that modifies the continuity and force equations, ultimately leading to density-gradient-sensitive corrections in the London equations and Meissner effect.

The Gist

Imagine you are trying to describe how a fluid, like water or air, moves. In the standard way of looking at quantum mechanics (the physics of the very small), scientists use a method called the Madelung transformation. Think of this like describing a river by looking at two separate things:

1. How deep the water is (the density).
2. The direction the current is flowing (the phase).

In the traditional view, these two things are independent. The depth of the water doesn't change how the current flows; it just sits there while the current moves based on the slope of the riverbed. The "current" is driven entirely by the slope (the phase), and the depth is just a passive passenger.

The New Idea: A "Squeezed" River
This paper proposes a different way to look at the quantum river. The author suggests that the depth of the water and the direction of the current are actually tightly linked in a specific, mathematical way.

Instead of a simple river, imagine a river where the water is made of a special, stretchy material. If the water gets deeper in one spot, it doesn't just sit there; it physically pulls and twists the direction of the current. The author calls this the "coth-Madelung ansatz."

Here is the core analogy:

• Standard View: The current is like a train on a track. The track (phase) decides where the train goes. The passengers (density) just sit there.
• This Paper's View: The train is made of the passengers. If the passengers crowd together (density increases), they physically reshape the track, forcing the train to change direction or speed up, even if the original track layout didn't change.

What This Changes

1. The Current Has a "Memory" of Density
In this new model, the speed of the quantum fluid isn't just determined by the slope of the track. It also depends on how quickly the "depth" of the fluid is changing.

• Analogy: Imagine walking through a crowd. In the old model, you walk based on the path ahead. In this new model, if the crowd gets denser right in front of you, you instinctively speed up or slow down because of that density, not just because of the path. The paper claims this creates a "density-gradient contribution" to the flow.

2. Superconductors Get "Textured"
The paper applies this idea to superconductors (materials that conduct electricity with zero resistance).

• Old View: Superconductors push magnetic fields out in a uniform, smooth way (the Meissner effect), like a perfect shield.
• New View: Because the "depth" of the superconductor's fluid affects the flow, the way it pushes out magnetic fields becomes patchy and textured. If the material has bumps or uneven density, the magnetic shielding changes shape to match those bumps. It's no longer a perfect, uniform shield; it's a flexible, adaptive one.

3. The "Zero Current" Trick
One of the most interesting findings is a special state where the electric current stops, even though there is a magnetic field and the material is uneven.

• Analogy: Imagine a river flowing against a strong wind. Usually, the wind stops the river. But in this new model, the river can "bend" its own path (change its internal shape) so perfectly that the wind's push is exactly canceled out by the river's new shape. The water stops moving, not because it's frozen, but because the internal geometry of the water has rearranged itself to balance the forces.

4. It Works Like a "Cole-Hopf" Transformation
The paper mentions that this math acts like a "generalized quantum Cole-Hopf transformation."

• Analogy: Think of a complex, messy knot of string (the standard quantum equations). This new math is like a special tool that untangles the knot, revealing that the messy parts were actually just a simple, smooth curve all along, but viewed through a "squeezed" lens. It simplifies the math of how the fluid accelerates by locking the speed directly to the shape of the density.

Summary
The paper argues that we have been treating the "amount" of a quantum particle (density) and its "direction" (phase) as separate things. The author suggests they are actually entangled.

By using a specific mathematical formula involving a hyperbolic function (coth), the author shows that the density of the quantum fluid actively shapes how it moves. This leads to a picture of quantum fluids and superconductors that are geometrically adaptive—they don't just flow; they reshape their own flow paths based on where the particles are crowded or sparse.

The paper does not claim this is a new law of nature that replaces everything we know, but rather a new mathematical lens that might explain complex behaviors in materials where density and flow are deeply mixed, such as in disordered superconductors or specific quantum tunneling scenarios.

Technical Summary

Technical Summary: Amplitude-Dependent Quantum Hydrodynamics from a coth-Madelung Ansatz

Problem Statement
Conventional quantum hydrodynamics relies on the Madelung transformation, which decomposes the wavefunction $\Psi$ into an amplitude (density) $\sqrt{\rho}$ and a phase $S/\hbar$ via the polar form $\Psi = \sqrt{\rho} e^{iS/\hbar}$. In this standard framework, amplitude and phase are treated as kinematically independent variables; the fluid velocity is determined solely by the phase gradient ($\mathbf{v} = \nabla S/m$), while the amplitude merely modulates the density. The author argues that this separation is a restrictive assumption that may not hold in strongly correlated, spatially structured, or highly disordered quantum systems (e.g., granular superconductors, pair-density-wave states) where density, coherence, and topology are deeply entangled. Furthermore, recent experimental discrepancies regarding tunneling particle speeds in evanescent fields suggest that the standard Bohmian guiding equation may be insufficient in certain regimes. The paper seeks to explore a formulation where the velocity field depends explicitly on both the phase and the amplitude of the wavefunction.

Methodology
The core methodology involves introducing a nonlinear, singular hyperbolic substitution for the wavefunction, termed the "coth-Madelung ansatz":
$$\Psi(\mathbf{r}, t) = R(\mathbf{r}, t) e^{i\theta(\mathbf{r}, t) \coth R(\mathbf{r}, t)}$$
where $R$ is a real amplitude field and $\theta$ is an auxiliary phase coordinate. Unlike the standard polar decomposition, the physical phase is defined as the composite quantity $\phi = \theta \coth R$.

The author proceeds by:

1. Deriving Kinematics: Calculating the probability current $\mathbf{j}$ and the resulting velocity field $\mathbf{v} = \mathbf{j}/\rho$. This reveals that $\mathbf{v}$ contains explicit terms dependent on the gradient of the amplitude ($\nabla R$), not just the phase gradient.
2. Formulating Dynamics: Substituting the ansatz into the Schrödinger equation to derive generalized continuity and Hamilton-Jacobi equations. The author also extends this formalism to the Gross-Pitaevskii equation (for Bose-Einstein condensates) and the Klein-Gordon equation.
3. Analyzing Consistency: Investigating the relationship between the generalized flow and the standard Madelung flow by analyzing the difference field $\Delta \mathbf{v}$. This involves deriving a nonlinear transport equation for the velocity difference and establishing kinematic compatibility conditions.
4. Applying to Electrodynamics: Reinterpreting $\Psi$ as a charged macroscopic order parameter (Ginzburg-Landau context) to derive modified London equations and analyze flux quantization.

Key Contributions and Results

• Amplitude-Dependent Velocity: The primary result is the derivation of a velocity field (Eq. 7) that includes a density-gradient contribution:
  $$\mathbf{v} = \frac{\hbar}{m} \coth R \left( \nabla \theta - \frac{2\theta \nabla R}{\sinh 2R} \right)$$
  This implies that local density textures actively participate in the kinematics of the particle, creating an "intrinsic internal shear" where density transport is self-structured by the order parameter's geometry.

• Generalized Hydrodynamic Equations: The continuity equation retains its standard form but acquires a different physical interpretation due to the velocity dependence on $\nabla R$. The Hamilton-Jacobi equation is modified such that the kinetic energy term $v^2$ contains mixed amplitude-phase terms and purely geometric amplitude-gradient terms. The quantum potential $Q$ remains unchanged in form, but the dynamics are richer due to the modified kinetic term.

• Scale-Invariant Structure: The paper demonstrates that the hyperbolic ansatz acts as a generalized quantum Cole-Hopf transformation. In specific limits (e.g., uniform phase), the fractional change in velocity becomes strictly scale-invariant and locked to the intrinsic curvature of the density envelope, effectively cleansing the kinematics of non-local, scale-dependent complexities associated with the traditional quantum potential.

• Modified Superconducting Electrodynamics: When applied to superconductivity, the framework yields generalized London equations.

  • First London Equation: The time derivative of the current includes a convective term proportional to $(\dot{\rho}/\rho)\mathbf{j}$ and a geometric acceleration term involving $\partial_t(\theta \coth R)$.
  • Second London Equation: The curl of the current ($\nabla \times \mathbf{j}$) acquires additional terms dependent on $\nabla \rho \times \nabla(\theta \coth R)$. Consequently, the Meissner effect is no longer described by a uniform screening length; magnetic field screening becomes sensitive to local amplitude textures and the relative orientation of density and phase gradients.

• Topological and Flux Modifications: The condition for single-valuedness applies to the composite field $\theta \coth R$, not $\theta$ alone. This leads to modified flux quantization rules where topological circulation can be redistributed among ordinary phase accumulation, amplitude-induced geometric twisting, and electromagnetic flux. The paper identifies "force-free compensation states" where the internal geometric momentum ($\hbar \nabla(\theta \coth R)$) exactly balances the gauge momentum ($q\mathbf{A}$), allowing for stationary states with zero current despite non-uniform amplitudes.

• Extensions to Other Equations: The formalism is successfully applied to the Gross-Pitaevskii equation (yielding modified dispersion relations for collective modes) and the Klein-Gordon equation. In the latter, the amplitude's temporal modulation acts as an active energy-shifting potential, altering the effective local rest mass of the field.

Significance and Claims
The paper claims that the coth-Madelung ansatz provides a "generalized hydrodynamic representation" where amplitude and phase are intrinsically coupled. The significance lies in shifting the understanding of the density field from a passive scalar that weights the current to a "genuinely geometric degree of freedom" that shapes the manifold along which current flows.

The author posits that this framework offers a natural route for modeling condensates where transport, screening, and topological structure are controlled by the geometry of the order parameter itself. Specifically, the paper suggests:

1. The Meissner effect should be spatially textured in inhomogeneous superconductors.
2. Amplitude collective modes may acquire direct electrodynamic relevance.
3. Topological defects (vortices) may exhibit internal amplitude-phase structures (e.g., split-core or annular textures) distinct from standard Ginzburg-Landau theory.
4. The framework may offer a mechanism to resolve issues in quantum tunneling (such as the Hartman effect) by providing a non-vanishing, amplitude-sensitive velocity field within classically forbidden regions.

The author maintains a modest stance, noting that it remains an open question whether this framework describes a deeper layer of quantum theory, an effective phenomenology for specific regimes, or a mathematically suggestive extension. However, the appearance of hyperbolic statistical kernels and amplitude-sensitive transport is presented as a compelling avenue for further investigation into the foundations of quantum hydrodynamics.