The Stationary Klein-Gordon Equation with a Delta-like Source: A Generalized Function Approach

TL;DR

This paper introduces a generalized function method to solve the stationary Klein-Gordon equation with a point-like source, providing a regular, unified solution that could simplify divergence handling in field theories.

Contribution

It presents a novel generalized function approach for solving the Klein-Gordon equation with singular sources, avoiding piecewise solutions and aiding in regularization and renormalization.

Findings

Solution is regular at the source singularity

Provides a single expression for the solution

Potentially simplifies renormalization in field theories

Abstract

This work aims to initiate a discussion on finding solutions to non-homoge\-neous differential equations in terms of generalized functions. For simplicity, we conduct the analysis within the specific context of the stationary Klein-Gordon equation with a point-like source, identifying a generalized function that solves such an equation and aligns with the solution obtained through the Fourier approach with dimensional regularization. In addition to being regular at the source singularity, a notable advantage of our solution is its presentation as a single expression, eliminating the need for piecewise definitions. The arguments presented here are applicable to a broader range of situations, offering a novel approach to addressing divergences in field theories using generalized functions. Moreover, we anticipate that the approach introduced in this work could provide a new method for handling Green functions regularized at coincident points, thereby simplifying the renorma\-lization process in a wide range of theories.