Uncertainties in the Foldy-Wouthuysen transformation

The Foldy-Wouthuysen (W-F) transformation is designed to transform the Hamiltonian of a fermion to a block diagonalized one in order to make the physical contents explicit. At the limit of zero interaction, this diagonalization can be done completely. With interactions. the diagonalization is incomplete and can be done only in the nonrelativistic limit to the extent of neglecting higher orders of 1/m where m is the mass of the fermion. Although the F-W transformations are not uniquely defined in either case, the ambiguity in the final results appears only in the case of incomplete diagonalization. For some very special interactions, this ambiguity has been studied, however, this article will show that the ambiguity exists in general. Therefore, care must be taken when displaying the physical contents of an interacting Hamiltonian by F-W transformation. Extra information must be invoked to guarantee the unique and correct result