Lessons from the Klein paradox

TL;DR

This paper re-examines the Klein paradox using quantum field theory, analyzing particle currents induced by strong electric potentials in 1+1 dimensions from a many-particle perspective.

Contribution

It provides a detailed quantum field theoretic analysis of the Klein paradox, exploring different potential scenarios and their effects on particle currents, with new insights into the physical interpretation.

Findings

Different mode bases yield zero or nonzero currents.

Rapid switching of potentials recovers standard current asymptotically.

Finite-duration potentials produce the standard current.

Abstract

We re-examine the Klein paradox from a many-particle perspective in quantum field theory. Specifically, we compute the expectation value of the particle current induced by a sufficiently strong step-like electric potential in 1+1 dimensions. First, for a constant (eternal) potential, we calculate the current for different Fock space ground states corresponding to distinct mode bases. While one basis yields a zero current, another produces the standard nonzero value. We then consider a potential that is rapidly switched on, recovering the standard current in the asymptotic future. This result is generalized to potentials that interpolate between different constant values at spatial infinity. Finally, we analyze a potential acting for a finite duration and again reproduce the standard current. A physical interpretation of these results is provided.