Foldy-Wouthuysen transformations in an indefinite-metric space. III. Relation to Lorentz transformations for first-order wave equations and the Poincaré generators

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Abstract
We show that there exists a definite relationship between a Lorentz transformation and a Foldy-Wouthuysen (FW) transformation for any relativistic wave equation in an indefinite-metric space which satisfies the following criteria: (i) The equation is first order with no external constraint equations. (ii) An adjoint equation (or, equivalently, a parity operator) exists. (iii) Lorentz transformation operators and related Poincaré generators are well defined. (iv) Any built-in subsidiary components can be decoupled. Our result allows us to obtain the explicit forms of the FW-transformed Poincaré generators from the original generators and in principle allows us to determine the exact, closed-form FW transformation.