Hamilton-Jacobi approach to Potential Functions in Information Geometry

TL;DR

This paper formulates the problem of finding potential functions for metric and tensor reconstruction as a Hamilton-Jacobi problem, linking it to quantum state geometry and classical information geometry contrast functions.

Contribution

It introduces a Hamilton-Jacobi framework for potential functions in information geometry, connecting quantum and classical geometric structures.

Findings

Establishes a Hamilton-Jacobi formulation for potential functions.

Links geometric structures in quantum mechanics and classical information geometry.

Provides a unified perspective on metric and tensor reconstruction.

Abstract

The search for a potential function $S$ allowing to reconstruct a given metric tensor $g$ and a given symmetric covariant tensor $T$ on a manifold $\mathcal{M}$ is formulated as the Hamilton-Jacobi problem associated with a canonically defined Lagrangian on $T\mathcal{M}$. The connection between this problem, the geometric structure of the space of pure states of quantum mechanics, and the theory of contrast functions of classical information geometry is outlined.