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.

The hyperplane formalism of Fleming is developed to include a discussion of the operations of the Poincaré group, as seen by an arbitrary hyperplane observer. Basis states for the m > 0 irreducible representations of the Poincaré group are re-expressed within the framework of generalized covariance provided by the hyperplane formalism and are seen to be related to the conventional helicity states by a special Lorentz transformation.

No abstract

The localized states of Newton and Wigner are reconstructed as a superposition of the canonical states of Foldy, and these states are then generalized to an arbitrary hyperplane via a procedure similar to that used in a previous work for the hyperplane generalization of helicity states. The corresponding hyperplane position operator, with mutually commuting components, is constructed and is seen to be equivalent to that local position operator given by Fleming.

Two-particle angular momentum states are constructed which are localized with respect to the magnitude of the relative position in the rest system and which have arbitrary 3-momentum dependence. The associated relative position operator is constructed, and a quantum-mechanical analog of the classical impact parameter is identified. Two-particle angular momentum states are constructed, which are also localized with respect to the ``mean-position'' of the 2-particle system, and the associated ``mean-position'' operator is seen to be a generalization of the 1-particle Newton-Wigner position operator.