Foldy-Wouthuysen transformations in an indefinite-metric space. I. Necessary and sufficient conditions for existence

Abstract

We prove that the necessary and sufficient conditions that a pseudounitary (Foldy-Wouthuysen) transformation exists which will diagonalize a nondiagonal pseudo-Hermitian matrix $\theta$ on a (nonsingular) indefinite-metric space are that all the eigenvalues of $\theta$ be real and all the eigenvectors of $\theta$ have nonzero norm. Physical applications are discussed. For example, the 2 × 2 case is discussed in general and for the Sakata-Taketani spin-0 field and the Lee model. This theorem also allows one to show that one can transform all the Bhabha Poincaré generators to a form which decouples the different mass (and normed) states.