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

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.