Unified Unitary Representation of the Poincaré Group for Particles of Zero and Positive Rest Mass

Abstract

As is well known, Wigner's construction of the unitary representations in terms of little groups gives quite dissimilar forms for the cases m = 0 and m > 0, while the spinor representation which does give a unified description of the above two cases is not unitary. We point out that it is quite possible to give a unified unitary representation for both cases (Sec. 1). This is achieved simply by noting that, while the factorization of U(Λ) corresponding the choice of the little group of (1, 0) is not valid for m = 0, the explicit final expression for the Wigner rotation R w and the corresponding infinitesimal generators remain perfectly well defined for m = 0, and continue to furnish a unitary (discrete spin) representation for this case, which is compatible with the restriction of helicity to a particular fixed value. Moreover, the representation thus obtained has a very simple and direct geometrical significance. The relation of our formulation with that of Wigner is studied (in Sec. 2) and the comparison with the spinor representations is given (Sec. 3). We rederive our representation, starting from a particular simple condition (4.1) (in Sec. 4) which holds for both cases (m = 0, m > 0). We then consider (Sec. 5) the application of our unified formulation to the reduction of direct products, involving particles of positive and zero rest mass, comparing the result with that of helicity coupling. In the Appendix we make certain remarks concerning the Hermiticity of the generators and the possibility of defining a position operator for m = 0. Comparison with Foldy's representation is given (in the Appendix and Sec. 3), explaining why his representation cannot be considered to be strictly unitary, though the relation with the unitary case is quite a simple one.