Comparative analysis of direct and "step-by-step" Foldy-Wouthuysen transformation methods

TL;DR

This paper compares direct and step-by-step Foldy-Wouthuysen transformation methods, highlighting their differences in higher-order terms and analyzing their applicability boundaries.

Contribution

It provides a detailed comparison of transformation methods, revealing discrepancies at higher orders and clarifying their respective applicability limits.

Findings

Step-by-step methods differ from the exact Eriksen method at higher orders.

Methods agree at zeroth and first orders but diverge thereafter.

The paper establishes boundaries for the applicability of each method.

Abstract

Relativistic methods for the Foldy-Wouthuysen transformation of the ``step-by-step'' type already at the first step give an expression for the Hamilton operator not coinciding with the exact result determined by the Eriksen method. The methods agree for the zeroth and first orders in the Planck constant terms but do not agree for the second and higher-order terms. We analyze the benefits and drawbacks of various methods and establish their applicability boundaries.