Poincaré-Irreducible Tensor Operators for Positive-Mass One-Particle States. I

Abstract

The main objective of this article is the relativistic generalization of the ordinary SO(3)-irreducible spin tensor operators for particles with positive mass. Two classes of relativistic one-particle tensor operators are constructed. The tensor operators of the first class transform according to those representations of the Poincaré group that are induced by the one-valued unitary irreducible representations of the pseudo-unitary group SU(1, 1) which belong to the continuous principal and the discrete principal series. These tensors are operator-valued functions of a spacelike 4-momentum transfer. The tensor operators of the second class correspond to vanishing 4-momentum transfer. They transform according to those representations of the Poincaré group that are induced by the unitary irreducible representations of the pseudo-orthogonal group SO(3, 1) or its universal covering group SL(2C) which belong to the principal series. Both classes of Poincaré-irreducible tensor operators are constructed in a spin helicity basis for timelike 4-momentum by means of projection operators which are continuous linear superpositions of unitary operator realizations for the groups SU(1, 1) and SL(2C). The Clebsch-Gordan coefficients associated with the reduction into the two classes of Poincaré-irreducible tensor operators of a dyadic product of spin-helicity basis vectors are calculated.