A review of the Klein paradox and its solution through quantum field theory - Guillermo Morfín

The Klein paradox exemplifies the challenges encountered in
developing a relativistic quantum mechanics framework. The paradox
arises when attempting to describe the behavior of a relativistic
particle using the methods of quantum mechanics. In this review, we
will explore how, by comparing the standard quantum mechanical
solution for the potential step with the solution for the Klein-Gordon
equation for the same potential, a significant issue arises in the
probabilities.
Finally, we will demonstrate how the problem is resolved by
considering that relativistic particles, upon interacting with the
potential, generate particle-antiparticle pairs.