[Paper Review] Information geometry and the hydrodynamical formulation of quantum mechanics

Mathieu Molitor|arXiv (Cornell University)|Apr 3, 2012

Quantum Mechanics and Applications5 references3 citations

TL;DR

This paper establishes an infinite-dimensional information geometry framework for quantum mechanics by endowing the space of smooth probability densities on a compact Riemannian manifold with a Fisher-like Riemannian metric and exponential-type affine connection. Using Dombrowski’s construction, it derives a symplectic structure on the tangent bundle that reproduces the Schrödinger equation, linking quantum dynamics to information geometry via hydrodynamical formulations.

ABSTRACT

Let (M,g) be a compact, connected and oriented Riemannian manifold. We denote D the space of smooth probability density functions on M. In this paper, we show that the Frechet manifold D is equipped with a Riemannian metric g^{D} and an affine connection abla^{D} which are infinite dimensional analogues of the Fisher metric and exponential connection in the context of information geometry. More precisely, we use Dombrowski's construction together with the couple (g^{D}, abla^{D}) to get a (non-integrable) almost Hermitian structure on D, and we show that the corresponding fundamental 2-form is a symplectic form from which it is possible to recover the usual Schrodinger equation for a quantum particle living in M. These results echo a recent paper of the author where it is stressed that the Fisher metric and exponential connection are related (via Dombrowski's construction) to Kahler geometry and quantum mechanics in finite dimension.

Motivation & Objective

• To extend finite-dimensional information geometry structures—specifically the Fisher metric and exponential connection—to infinite-dimensional spaces of smooth probability densities on a Riemannian manifold.
• To investigate whether the hydrodynamical formulation of quantum mechanics can be derived from geometric structures in information geometry.
• To establish a correspondence between the Schrödinger equation and a symplectic form arising from Dombrowski’s construction on the tangent bundle of the statistical manifold of probability densities.
• To demonstrate that wave functions in quantum mechanics emerge naturally from the geometric structure of evolving probability densities, generalizing finite-dimensional results.

Proposed method

• Define the Fréchet manifold $\mathcal{D}$ of smooth, positive probability densities on a compact Riemannian manifold $(M,g)$ with unit total measure.
• Construct an infinite-dimensional analogue of the Fisher metric $g^{\mathcal{D}}$ and exponential connection $\nabla^{\mathcal{D}}$ on $\mathcal{D}$ using Dombrowski’s construction.
• Apply Dombrowski’s method to the pair $(g^{\mathcal{D}}, \nabla^{\mathcal{D}})$ to obtain an almost Hermitian structure on the tangent bundle $T\mathcal{D}$.
• Show that the fundamental 2-form of this almost Hermitian structure is symplectic and governs the dynamics of the system.
• Derive the Schrödinger equation from this symplectic structure by identifying the Hamiltonian vector field associated with the energy functional.
• Construct a wave function $\Psi = \sqrt{\rho} e^{-i\phi/\hbar}$ from the hydrodynamical decomposition $\rho = |\psi|^2$, $\phi = \text{Im}(\log \psi)$, linking probability densities to quantum states.

Experimental results

Research questions

• RQ1 Can the Fisher metric and exponential connection be generalized to infinite-dimensional spaces of smooth probability densities on a Riemannian manifold?
• RQ2 Does Dombrowski’s construction on this infinite-dimensional statistical manifold yield a symplectic structure that reproduces the Schrödinger equation?
• RQ3 How is the wave function in quantum mechanics related to the geometric structure of evolving probability densities in information geometry?
• RQ4 Is the hydrodynamical formulation of quantum mechanics derivable from information-geometric principles in infinite dimensions?
• RQ5 Can the standard quantum formalism emerge from purely statistical and geometric foundations via this construction?

Key findings

• The space $\mathcal{D}$ of smooth probability densities on a compact Riemannian manifold $(M,g)$ is equipped with a Riemannian metric $g^{\mathcal{D}}$ and affine connection $\nabla^{\mathcal{D}}$ that are infinite-dimensional analogues of the Fisher metric and exponential connection.
• Dombrowski’s construction applied to $(g^{\mathcal{D}}, \nabla^{\mathcal{D}})$ yields a non-integrable almost Hermitian structure on $T\mathcal{D}$ with a fundamental 2-form that is symplectic.
• This symplectic form on $T\mathcal{D}$ generates a Hamiltonian flow that reproduces the Schrödinger equation for a quantum particle on $M$ with potential $V$.
• The wave function $\Psi = \sqrt{\rho} e^{-i\phi/\hbar}$, derived from the hydrodynamical decomposition of the density $\rho$, provides a natural embedding of the statistical manifold into a Hilbert space of complex-valued $L^2$ functions.
• The construction generalizes finite-dimensional results: the Kähler structure on $T\mathcal{P}_n^\times$ via Dombrowski’s method is isomorphic to $\mathbb{P}(\mathbb{C}^n)$, and the present work extends this to infinite dimensions.
• The paper establishes a geometric derivation of quantum mechanics from information-geometric principles, suggesting that wave functions and Hermitian operators arise naturally from statistical structures on probability densities.