Bhabha first-order wave equations. VI. Exact, closed-form, Foldy-Wouthuysen transformations and solutions

We give a closed-form, finite-polynomial expression for the Foldy-Wouthuysen (FW) transformation $U^{-1\mathrm{}}$ for arbitrary-spin Bhabha fields. Our result is obtained by appropriately normalizing the Lorentz transformation operator, expressing this transformation as a finite polynomial, and then properly interpreting the energy, mass, and especially momentum operators involved. For integer-spin fields the built-in subsidiary components are projected out. An algorithm is given which allows one to easily write the expression for the FW transformation of any Bhabha field. We comment on the properties of $U^{-1\mathrm{}}$. We note that the columns of the FW transformation are the metric-orthonormal eigenvectors of the Hamiltonian, ${\hat{u}}_k$, and provide the relation of the ${\hat{u}}_k$ to the solutions of the wave equation, ${\psi }_k$. Special cases up to $\mathcal{S} = 3$ are listed and investigated. Some physical and mathematical applications of our method and results are also given.