Perturbative quantum analysis and classical limit of the electron scattering by a solenoidal magnetic field

TL;DR

This paper investigates the differences between classical and quantum scattering of electrons by a magnetic field in a solenoid, revealing that classical behavior cannot be recovered from perturbative quantum results, suggesting the need for non-perturbative methods.

Contribution

The study demonstrates that perturbative quantum calculations do not reproduce classical scattering results, highlighting the necessity of non-perturbative approaches for accurate classical-quantum correspondence.

Findings

Classical and quantum differential cross sections differ significantly.

Higher order quantum corrections do not recover the classical result.

Non-perturbative methods are required for classical limit recovery.

Abstract

A well known example in quantum electrodynamics (QED) shows that Coulomb scattering of unpolarized electrons, calculated to lowest order in perturbation theory, yields a results that exactly coincides (in the non-relativistic limit) with the Rutherford formula. We examine an analogous example, the classical and perturbative quantum scattering of an electron by a magnetic field confined in an infinite solenoid of finite radius. The results obtained for the classical and the quantum differential cross sections display marked differences. While this may not be a complete surprise, one should expect to recover the classical expression by applying the classical limit to the quantum result. This turn not to be the case. Surprisingly enough, it is shown that the classical result can not be recuperated even if higher order corrections are included. To recover the classic correspondence of the quantum scattering problem a suitable non-perturbative methodology should be applied.