Foldy-Wouthuysen transformations in an indefinite-metric space. IV. Exact, closed-form expressions for first-order wave equations

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Abstract
We use a powerful matrix theorem to derive the closed-form, finite-polynomial, matrix expression of a Lorentz transformation for the class of relativistic, first-order wave equations described in the preceding paper. Combined with the method developed there, this theorem allows the exact, closed-form expression to be given for a Foldy-Wouthuysen (FW) transformation. An algorithm is given which is a simple procedural prescription for writing down the FW transformation results we have obtained. We discuss two specific examples which illustrate our results, and in a third example show that our method must be modified if the wave equation is not first-order in space.