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Physical Review

Relationship of the Foldy-Wouthuysen Transformation to Lorentz Transformations

Herbert Jehle, William C. Parke

Published Feb 8, 1965• 14 citations
Publisher

Abstract

The covariant spinor formalism of Infeld and van der Waerden and of Uhlenbeck and Laporte permits a simple interpretation of the relationship between a Foldy-Wouthuysen transformation for a free particle and an associated pair of Lorentz transformations.
DOI
10.1103/physrev.137.b760

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Physical Review•Nov 15, 1963•77 citations
An examination is made of the consequences for the quantum mechanics of spinning particles of equations characteristic of Lorentz-covariant position variables. These equations are commutator analogs of the Poisson bracket equations that express the familiar transformation properties of space-time events in classical mechanics. For a particle of zero spin it is found that the usual canonical coordinate is the unique solution of these equations. For a particle with positive spin there is no position operator which satisfies these equations and has commuting components. For a particle and antiparticle there is a unique solution with commuting components which is valid for all values of the spin and reduces for zero spin to the canonical coordinate. For spin \textonehalf{} this is the Foldy-Wouthuysen transform of the position operator of the Dirac equation. A generalization of the inverse Foldy-Wouthuysen transformation, valid for any value of the spin, appears as a unique unitary transformation which takes this generalized Dirac position to the canonical coordinate. The application of this transformation to the canonical form of the Hamiltonian gives a generalization of the Dirac equation Hamiltonian. This is developed and compared with the literature for spin 1. It gives a nonlocal equation as the spin 1 analog of the Dirac equation.