Canonical Form of the Covariant Free-Particle Equations

Abstract

The canonical form of the covariant equations for free particles of nonzero rest mass is proposed to be taken as [(p νpν)12−κ] ψ=0, instead of [γ0(p2+κ2)12−p0] ψ=0, as suggested by Foldy. The connection of our representation with the usual forms of the Dirac and the Klein-Gordon (K-G) equations are discussed, each feature being compared with the corresponding one in Foldy's case. The case of the Dirac equation is treated in some detail. A study of the infinitesimal operators of the Poincaré group and the transformation properties of the wavefunction and the polarization operator in our representation lead us to conclude that the choice of operators and the definition of spin states adopted by Iu. M. Shirokov in his study of the Poincaré group corresponds directly to our representation and the canonical form proposed, rather than that proposed by Foldy, as is sometimes supposed. It is also shown that the proposed canonical form corresponds to Wigner's unitary representation of the Poincaré group in terms of the little group of (κ, 0, 0, 0) (for κ > 0). In Appendix A, we give a brief outline of the decomposition of the direct-product representation of the Poincaré group to bring out the special features that arise in our representation. In Appendix B, we compare in detail, for the case of the Dirac equation, our transformation with the well-known Foldy-Wouthuysen transformation. The case of zero rest mass has not been considered. Also the discussion of the position operators in our representation has been left aside and is to be taken up in a following article.