Reduction of Reducible Representations of the Poincaré Group to Standard Helicity Representations

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Abstract
In this paper we introduce realizations of the generators of the Poincaré group for real and imaginary masses which are close in form to the Lomont‐Moses realizations for zero mass. These realizations (which we call ``standard helicity realizations or representations'') are characterized by the way that the infinitesimal generators are given in terms of the helicity operator. We also give the global form of the realizations and discuss in detail the realizations for the case that they are unitary and irreducible. We then show how any reducible representation of the Poincaré group for which the infinitesimal generators of the translation and rotation subgroups are Hermitian and integrable and for which the space‐time generators are integrable (but not necessarily Hermitian) can be reduced to the standard helicity realizations. In the case that the reducible representation is unitary, this process enables one to reduce the reducible representation to irreducible unitary standard helicity representations. Finally, we show how the Foldy‐Shirokov realizations for real mass are related to the standard helicity representation.