Geometric Madelung Transform

• Geometric Madelung Transform is a method that converts quantum wavefunction dynamics into real-valued hydrodynamic variables via polar decomposition and symmetry reduction.
• It establishes a symplectomorphism and isometry between the quantum Hilbert space with the Fubini–Study metric and the cotangent bundle of probability densities with the Fisher–Rao metric.
• The framework unifies quantum mechanics and fluid dynamics, facilitating practical applications in molecular dynamics and quantum-classical hybrid models.

The geometric Madelung transform is a precise mathematical procedure that recasts quantum wavefunction dynamics, most notably the Schrödinger equation, into a real-valued, hydrodynamical framework using infinite-dimensional geometry, symplectic reduction, and variational principles. Beyond its classical polar decomposition, the geometric Madelung transform constitutes a symplectomorphism and isometry between the projective space of wave functions endowed with the Fubini-Study metric and the cotangent bundle to the space of probability densities with the Fisher-Rao metric. This framework interlinks quantum mechanics, hydrodynamics, information geometry, and geometric mechanics, underpinning both rigorous mathematical structures and practical reduction schemes for quantum dynamical systems, including those appearing in Born-Oppenheimer molecular dynamics and quantum-classical hybrid models (Bergold et al., 2023, Khesin et al., 2018).

1. Geometric Madelung Transform: Polar Decomposition and Symmetry Reduction

The geometric Madelung transform is a two-step process:

1. Polar decomposition: For a wave function $\psi(x,t)$ (e.g., solving the Schrödinger equation), write $\psi(x,t) = \sqrt{D(x,t)}\, e^{i S(x,t)/\hbar}$, where $D(x,t) = |\psi(x,t)|^2 \geq 0$ is the mass or probability density, and $S(x,t)$ is the real-valued Hamilton principal function.
2. Symmetry reduction: Map the Hilbert space to the cotangent bundle of densities and velocities using Hamilton's variational principle. The symmetry group is a semidirect product $$G = \mathrm{Diff}(\mathbb{R}^n) \ltimes F(\mathbb{R}^n,U(1))$$, combining diffeomorphisms (mass transport) and $U(1)$ phase rotations. The Dirac–Frenkel Lagrangian is invariant under this group, and the reduction yields Euler–Poincaré equations for $(D,u)$, where $u(x,t) = \frac{1}{m}\nabla S(x,t)$ is the hydrodynamic velocity (Bergold et al., 2023).

2. Kähler Geometry, Symplectomorphism, and Momentum Maps

The transform defines a Kähler map between the projective Hilbert space $P(H^s(M;\mathbb{C}))$ (with the Fubini–Study metric) and $T^*\mathrm{Dens}^s(M)$ (with the Fisher–Rao metric). The mapping

$\psi(x,t) = \sqrt{D(x,t)}\, e^{i S(x,t)/\hbar}$0

is a symplectomorphism: the pullback of the Fubini–Study symplectic form $\psi(x,t) = \sqrt{D(x,t)}\, e^{i S(x,t)/\hbar}$1 is, up to a constant, the canonical symplectic form $\psi(x,t) = \sqrt{D(x,t)}\, e^{i S(x,t)/\hbar}$2 on $\psi(x,t) = \sqrt{D(x,t)}\, e^{i S(x,t)/\hbar}$3:

$\psi(x,t) = \sqrt{D(x,t)}\, e^{i S(x,t)/\hbar}$4

Additionally, $\psi(x,t) = \sqrt{D(x,t)}\, e^{i S(x,t)/\hbar}$5 is an isometry, and these spaces are both Kähler: the complex structure $\psi(x,t) = \sqrt{D(x,t)}\, e^{i S(x,t)/\hbar}$6 on $\psi(x,t) = \sqrt{D(x,t)}\, e^{i S(x,t)/\hbar}$7 satisfies $\psi(x,t) = \sqrt{D(x,t)}\, e^{i S(x,t)/\hbar}$8, where $\psi(x,t) = \sqrt{D(x,t)}\, e^{i S(x,t)/\hbar}$9 is the Sasaki–Fisher–Rao metric (Khesin et al., 2018, Khesin et al., 2017). The Madelung transform also serves as a momentum map for the group action on wave functions, intertwining the Hamiltonian structure of quantum and hydrodynamic phase spaces (Fusca, 2015).

3. Hamiltonian and Hydrodynamical Structures

Reduction of the Dirac–Frenkel Lagrangian via the geometric Madelung transform yields a Lagrangian and Hamiltonian on the hydrodynamic variables:

$D(x,t) = |\psi(x,t)|^2 \geq 0$0

transforms to

$D(x,t) = |\psi(x,t)|^2 \geq 0$1

The resulting equations—the continuity and quantum Hamilton–Jacobi equations—together encode the original quantum evolution in a hydrodynamic form, closing via Hamilton’s equations for the canonical Poisson structure of the density-phase variables (Carles et al., 2011, Reddiger, 2015). The quantum potential,

$D(x,t) = |\psi(x,t)|^2 \geq 0$2

acts as an internal pressure or curvature term (Delphenich, 2013).

4. Applications: Variational Asymptotics, Molecular Dynamics, and Mixed Quantum-Classical Systems

The geometric Madelung framework allows rigorous variational asymptotics. In Born–Oppenheimer molecular dynamics, one applies the transform to the nuclear factor in the exact-factorization $D(x,t) = |\psi(x,t)|^2 \geq 0$3. Upon taking the small electron-to-nuclear mass ratio limit in the reduced action, the equations reduce to Newton’s equations for the nuclei on the Born–Oppenheimer surface, aligning with the classical action

$D(x,t) = |\psi(x,t)|^2 \geq 0$4

Moreover, "bohmion" particle closure schemes arise by regularizing and discretizing the quantum pressure term, providing on-the-fly, finite-dimensional hydrodynamic models for BOMD (Bergold et al., 2023).

5. Generalizations and Geometric Analogies

The geometric Madelung transform is not restricted to standard quantum mechanics. It extends to nonlinear Schrödinger equations, barotropic Euler systems, shallow water models, and more. In one dimension, the Hasimoto transform mapping vortex filament dynamics to the nonlinear Schrödinger equation is a special case. Higher-dimensional analogues remain partly conjectural but are guided by conservation laws (e.g., Willmore energy) and the symplectic/Kähler structure (Khesin et al., 2018, Khesin et al., 2017).

Geometric interpretations of the quantum potential $D(x,t) = |\psi(x,t)|^2 \geq 0$5 yield two parallel pictures: - Scalar curvature (Levi-Civita): $D(x,t) = |\psi(x,t)|^2 \geq 0$6 is proportional to the scalar curvature of a conformal metric $D(x,t) = |\psi(x,t)|^2 \geq 0$7. - Teleparallel (frame-strain): $D(x,t) = |\psi(x,t)|^2 \geq 0$8 encodes the "strain" in a parallel frame field, leading to a constitutive law for Madelung stress analogous to metric elasticity or bending of elastic wires (Delphenich, 2013).

6. Significance: Unification, Symplectic Realization, and Information Geometry

The geometric Madelung formalism unifies quantum dynamics and classical fluid mechanics under a single variational, symplectic, and information-geometric paradigm. It forms an explicit isomorphism between quantum and hydrodynamical phase spaces, enables reduction by infinite-dimensional symmetry groups, and places quantum evolution within a Hamiltonian system on $D(x,t) = |\psi(x,t)|^2 \geq 0$9 equipped with the Fisher–Rao metric.

The transform identifies canonical "Clebsch variables" for hydrodynamics, serves as a symplectic realization of the Lie–Poisson structure of compressible Euler equations, and leverages the optimal-transport (Wasserstein) geometry for analysis of quantum-to-classical limits and singular solutions (e.g., vortex filaments) (Fusca, 2015, Khesin et al., 2020). These connections position the geometric Madelung transform at the crossroads of quantum mechanics, fluid dynamics, and information geometry, with broad impact on the analysis of variational limits, mixed quantum–classical dynamics, and the geometric structure of quantum theories.