Hamilton-Jacobi Theory and Information Geometry

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.

8 pages

References in corpus (1)

• From tools in symplectic and Poisson geometry to Souriau's theories of statistical mechanics and thermodynamics

Cited by in corpus (3)

• Hamilton-Jacobi theory and integrability for autonomous and non-autonomous contact systems
• Generalised Potential Functions in Differential Geometry and Information Geometry
• Lagrangian description of Heisenberg and Landau-von Neumann equations of motion