Wave Equations on a Hyperplane

Abstract

The Dirac equation, the Weaver-Hammer-Good wave equations, and the Weinberg wave equations are written in a manifestly covariant form in terms of hyperplane parameters according to Fleming's hyperplane formalism. A Fody-Wouthuysen-type wave equation is developed for the Weinberg theory and it, along with the usual Foldy-Wouthuysen wave equation and transformation, is also written in a manifestly covariant form for all spin. Fleming's formalism is extended to include the case where the hyperplane parameters are operators as well as $c$ numbers. As a consequence, a hyperplane observer which corresponds to the particle rest frame is considered, with the result that wave equations are obtained in the usual manifestly covariant form for all spin with no auxiliary conditions or unphysical solutions.