Notes On The Klein-Gordon Equation

TL;DR

This paper derives the Klein-Gordon equation using an information theory framework, connecting it to Fokker-Planck and Schrödinger equations, and explores extensions to curved space-time and non-linear dynamics.

Contribution

It introduces an information-theoretic derivation of the Klein-Gordon equation, incorporating potentials, non-extensive statistics, and curved space-time considerations.

Findings

Derived the Klein-Gordon equation from maximum entropy principles.

Established connections to Fokker-Planck and Schrödinger equations.

Compared results with path integral approaches in curved space-time.

Abstract

In this article, we derive the scalar parametrized Klein-Gordon equation from the formal information theory framework. The least biased probability distribution is obtained, and the scalar equation is recast in terms of a Fokker-Planck equation in terms of the imaginary time, or a Schroedinger equation for the proper time. This method yields the Green's function parametrized by an evolution parameter. The derivation can then allow the use of potentials as constraints along with the Hamiltonian or moments of the evolution. The information theoretic, analogously the maximum entropy method, also allows one to examine the possibility of utilizing generalized and non-extensive statistics in the derivation. This approach yields non-linear evolution in the parametrized Klein-Gordon partial differential equations. Furthermore, we examine the Klein-Gordon equation in curved space-time, and we compare our results to the results of Schwinger and Dewitt obtained from path integral approaches.