Non-locality of Foldy-Wouthuysen and related transformations for the Dirac equation

TL;DR

This paper investigates the non-local effects of Foldy-Wouthuysen and similar transformations in the Dirac equation, revealing that these transformations cause a smearing of functions over a scale comparable to the Compton wavelength.

Contribution

It provides a detailed analysis of the non-locality inherent in Foldy-Wouthuysen transformations, including calculations of second moments and variances of the transformed functions.

Findings

Transformed functions are smeared over the Compton wavelength scale.

All transformed quantities exhibit non-locality comparable to the Compton wavelength.

The study quantifies the non-local effects of these transformations in the Dirac equation.

Abstract

Non-localities of Foldy-Wouthuysen and related transformations, which are used to separate positive and negative energy states in the Dirac equation, are investigated. Second moments of functional kernels generated by the transformations are calculated, the transformed functions and their variances are computed. It is shown that all the transformed quantities are smeared in the coordinate space by the amount comparable to the Compton wavelength $\lambda_c=\hbar/mc$.