QFT Treatment of the Klein Paradox

TL;DR

This paper applies quantum field theory, specifically the path integral formalism, to analyze the Klein paradox, providing a more rigorous understanding of electron scattering and clarifying the paradox's origin.

Contribution

It introduces a quantum field theoretical approach to the Klein paradox, moving beyond single-particle quantum mechanics to evaluate reflection and transmission coefficients.

Findings

Path integral formalism yields reflection and transmission coefficients.

Clarifies the Klein paradox as a consequence of ill-defined step potential.

Provides a field-theoretic perspective on electron scattering.

Abstract

It is well known that, Klein paradox is one of the most exotic and counterintuitive consequences of quantum theory. Nevertheless, many discussions about the Klein paradox are based upon single-particle Dirac equation in quantum mechanics rather than quantum field method. By using the path integral formalism, we evaluate the reflection and transmission coefficients up to the lowest order for the electron scattering by the finite square barrier potential. Within the context of assuming the step potential is the limiting case of the finite square barrier potential, we explain the Klein paradox that is caused by the ill-definition of the step potential.