Klein paradox for bosons, wave packets and negative tunnelling times

TL;DR

This paper investigates the Klein paradox for bosons, revealing that bosons can seemingly cross barriers with negative tunneling times through acausal mechanisms, and explores how smooth barriers mitigate this paradox.

Contribution

It demonstrates the acausal nature of bosonic tunneling in the Klein paradox and shows how smooth barriers resolve the divergence issues present in rectangular barriers.

Findings

Bosons can cross supercritical barriers with negative tunneling times.

Acausal transmission is linked to divergent multiple reflections.

Smooth barriers prevent divergence and enhance pair production.

Abstract

We analyse a little known aspect of the Klein paradox. A Klein-Gordon boson appears to be able to cross a supercritical rectangular barrier without being reflected, while spending there a negative amount of time. The transmission mechanism is demonstrably acausal, yet an attempt to construct the corresponding causal solution of the Klein-Gordon equation fails. We relate the causal solution to a divergent multiple-reflections series, and show that the problem is remedied for a smooth barrier, where pair production at the energy equal to a half of the barrier's height is enhanced yet remains finite.