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

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.