Schroedinger revisited:How the time-dependent wave equation follows from the Hamilton-Jacobi equation

Abstract

It is shown how using the classical Hamilton-Jacobi equation one can arrive at the time-dependent wave equation. Although the former equation was originally used by E.Schroedinger to get the wave equation, we propose a different approach. In the first place, we do not use the principle of least action and, in addition, we arrive at the time-dependent equation, while Schroedinger (in his first seminal paper) used the least action principle and obtained the stationary wave equation. The proposed approach works for any classical Hamilton-Jacobi equation. In addition, by introducing information loss into the Hamilton-Jacobi equation we derive in an elementary fashion the wave equations (ranging from the Shroedinger to Klein-Gordon, to Dirac equations). We also apply this technique to a relativistic particle in the gravitational field and obtain the respective wave equation. All this supports 't Hooft's proposal about a possibility of arriving at quantum description from a classical continuum in the presence of information loss.