The Klein-Gordon's field. A counter-example of the classical limit

TL;DR

This paper investigates the Klein-Gordon field with a homogeneous external potential, revealing that particle-antiparticle pair creation phenomena vanish in one dimension as Planck's constant approaches zero, but persist in higher dimensions.

Contribution

It constructs the Fock space for the Klein-Gordon field with an external potential and demonstrates the dimensional dependence of pair creation phenomena in the classical limit.

Findings

Pair creation phenomena vanish in 1D as 

In 2D and 3D, pair creation probability remains non-zero as 

The classical limit behavior differs across spatial dimensions

Abstract

We will study the Klein-Gordon's field with an homogeneous external potential, which does not depend on $\h$. We will construct the Fock's space corresponding to our problem and we will see that there are phenomena of creation and anihilation of pairs particle-antiparticle. Finally, we will see that in dimension 1, when $\h\to 0$, these phenomena disappear. However, in dimension 2 or 3, when $\h\to 0$, the creation probability of particle-antiparticle pairs is not zero.