Virtual beams and the Klein paradox for the Klein-Gordon equation

TL;DR

This paper addresses the Klein paradox in the Klein-Gordon equation by introducing virtual beams and a generalized method of images, providing a clearer understanding without quantum field theory complexities.

Contribution

It proposes a novel solution to the Klein paradox for the Klein-Gordon equation using virtual beams and a generalized method of images, extending recent Dirac equation results.

Findings

A global reflection coefficient is derived from reflected and virtual particles.

The method offers a simple, effective understanding of the Klein paradox.

The approach avoids quantum field theory complexities.

Abstract

Whenever we consider any relativistic quantum wave equation we are confronted with the Klein paradox, which asserts that incident particles will suffer a surplus of reflection when dispersed by a discontinuous potential. Following recent results on the Dirac equation, we propose a solution to this paradox for the Klein-Gordon case by introducing virtual beams in a natural well-posed generalization of the method of images in the theory of partial differential equations. Thus, our solution considers a global reflection coefficient obtained from the two contributions, the reflected particles plus the incident virtual particles. Despite its simplicity, this method allows a reasonable understanding of the paradox within the context of the quantum relativistic theory of particles (according to the original setup for the Klein paradox) and without resorting to any quantum field theoretic issues.