A possibility of Klein Paradox in quaternionic (3+1) frame

TL;DR

This paper investigates the potential existence of the Klein paradox within a quaternionic (3+1)-dimensional framework, extending the Klein-Gordon equation to quaternionic fields and analyzing relativistic particle behavior.

Contribution

It introduces a quaternionic form of the Klein-Gordon equation and explores the conditions under which the Klein paradox occurs in this non-commutative setting.

Findings

Klein paradox occurs only in the Klein zone with specific energy conditions.

Quaternionic reflection coefficient exceeds one in the Klein paradox.

Quaternionic transmission coefficient drops below zero during the paradox.

Abstract

In light of the significance of non-commutative quaternionic algebra in modern physics, the current study proposes the existence of the Klein paradox in the quaternionic (3+1)-dimensional space-time structure. By introducing the quaternionic wave function, we rewrite the Klein-Gordon equation in an extended quaternionic form that includes scalar and vector fields. Because quaternionic fields are non-commutative, the quaternionic Klein-Gordon equation provides three separate sets of the probability density and probability current density of relativistic particles. We explore the significance of these probability densities by determining the reflection and transmission coefficients for the quaternionic relativistic step potential. Furthermore, we also discuss the quaternionic version of the oscillatory, tunnelling, and Klein zones for the quaternionic step potential. The Klein paradox occurs only in the Klein zone when the impacted particle's kinetic energy is less than \mathbb{V}_{0}-m_{0}c^{2}. Therefore, it is emphasized that for the quaternionic Klein paradox, the quaternionic reflection coefficient becomes exclusively higher than value one while the quaternionic transmission coefficient becomes lower than zero.