Vacuum decay and fermion total reflection by the Klein step

TL;DR

This paper reframes Klein tunneling as a quantum-field-theoretic scattering problem for a one-dimensional Dirac fermion on a step potential, focusing on asymptotic states at $t\to\pm\infty$ and defining two unstable vacua connected by a Bogoliubov-like transformation. In the Klein region ($m<E<V_0-m$), the field-theoretic treatment reveals vacuum decay with rate $\gamma$ and shows that an incident fermion is completely reflected ($P_r=1$) while vacuum-decay–generated particle–antiparticle pairs populate the out-state. The approach uses a complete set of stationary Dirac solutions to build a quantized field $\Psi(z,t)$, defines $\text{in}$ and $\text{out}$ operators, and employs generating functionals to quantify multipair production. The results illuminate the interplay between vacuum instability and relativistic scattering, offering a consistent picture of Klein tunneling beyond single-particle quantum mechanics and suggesting extensions to other mode structures.

Abstract

The so-called Klein tunneling is re-examined within the framework of quantum field theory, but from a different point of view on the asymptotic states. We treat it as a one-dimensional scattering process of a fermion incident to a step potential and introduce asymptotic operators as appropriate $t = \pm \infty$ limits of the field operator responsible for the process. For the so-called Klein energy range, two asymptotic vacua naturally emerge which are defined as states annihilated by the asymptotic annihilation operators. They are related by a similarity transformation, which entails a vacuum decay and yields a vacuum decay constant. When a fermion with incident energy in the Klein region is injected to the step, it is shown to be reflected with probability one, accompanied by fermion--anti-fermion pairs that are vacuum decay products.

Figures (1)

• Figure 1: Scattering states are categorized according to whether an incident flux is coming from the left ($\psi$) or right ($\phi$) of the step potential and to the range of their energies.