Some Aspects of the Exact Foldy-Wouthuysen Transformation for a Dirac Fermion

TL;DR

This paper explores the exact Foldy-Wouthuysen transformation for Dirac fermions, demonstrating how it can be applied even without an involution operator to recover standard results.

Contribution

It provides a detailed analysis of the exact FWT, including methods to obtain perturbative results without an involution operator, with illustrative examples.

Findings

Exact FWT can be applied without an involution operator.

The technique reproduces conventional perturbative results.

Several specific cases demonstrate the method's effectiveness.

Abstract

The Foldy-Wouthuysen transformation (FWT) is used to separate distinct components of relativistic spinor field, e.g. electron and positron. Usually, the FWT is perturbative, but in some cases there is an involution operator and the transformation can be done exactly. We consider some aspects of an exact FWT and show that, even if the theory does not admit an involution operator, one can use the technique of exact FWT to obtain the conventional perturbative result. Several particular cases can be elaborated as examples.