Hamilton-Jacobi Theory and Information Geometry

TL;DR

This paper reviews a Hamilton-Jacobi based method to define divergence functions on statistical manifolds and explores the inverse problem of deriving Lagrangians from known divergences, with applications to quantum systems.

Contribution

It introduces a framework for constructing divergence functions via Hamilton-Jacobi theory and investigates the inverse problem of identifying Lagrangians from divergences, extending to quantum contexts.

Findings

Reviewed the Hamilton-Jacobi construction of divergence functions.

Formulated the inverse problem for divergence functions and Lagrangians.

Discussed extension to quantum systems using probability amplitudes.

Abstract

Recently, a method to dynamically define a divergence function $D$ for a given statistical manifold $(\mathcal{M}\,,g\,,T)$ by means of the Hamilton-Jacobi theory associated with a suitable Lagrangian function $\mathfrak{L}$ on $T\mathcal{M}$ has been proposed. Here we will review this construction and lay the basis for an inverse problem where we assume the divergence function $D$ to be known and we look for a Lagrangian function $\mathfrak{L}$ for which $D$ is a complete solution of the associated Hamilton-Jacobi theory. To apply these ideas to quantum systems, we have to replace probability distributions with probability amplitudes.