Poincaré‐Irreducible Tensor Operators for Positive‐Mass One‐Particle States. II

TLDR: In this article, the relativistic generalization of the ordinary SO(3)-irreducible spin tensor operators for particles with positive mass is presented. But the main objective of this paper is to construct a spin helicity basis for spin tensors.

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 Poincare 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 Poincare 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 Poincare‐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 Poincare‐irreducible tensor operators of a dyadic product of spin‐helicity basis vectors are calculated.