The Hamilton--Jacobi Theory and the Analogy between Classical and Quantum Mechanics

TL;DR

This paper reviews various aspects of Hamilton-Jacobi theory, exploring its classical and quantum connections, including less conventional topics like tangent bundles, differential operators, and Lie groups.

Contribution

It provides a comprehensive overview of both conventional and novel applications of Hamilton-Jacobi theory in classical and quantum mechanics.

Findings

Connections between classical and quantum Hamilton-Jacobi theories clarified

Extensions of HJ theory to tangent bundles and Lie groups explored

Quantum HJ theory and differential operator problems discussed

Abstract

We review here some conventional as well as less conventional aspects of the time-independent and time-dependent Hamilton-Jacobi (HJ) theory and of its connections with Quantum Mechanics. Less conventional aspects involve the HJ theory on the tangent bundle of a configuration manifold, the quantum HJ theory, HJ problems for general differential operators and the HJ problem for Lie groups.