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

The present communication studies relations between the two classes of Poincaré-irreducible tensor operators which were constructed in a previous paper. The tensor operators of the first class transform according to those representations of the Poincaré group that are induced by the one-valued irreducible unitary representations of SU(1, 1) which belong to the continuous and the discrete principal series. The tensor operators of the second class transform according to the Poincaré group representations that are induced by the irreducible unitary principal series representations of SL(2, C). The Poincaré-irreducible tensor operators of the second class are decomposed into Lorentz-irreducible tensor operators which transform by the irreducible unitary SL(2, C) representations of the principal series. The expansions of a spin density matrix describing a statistical ensemble of wavepackets into Poincaré- and Lorentz-irreducible components are derived. The infinite-dimensional associative algebra for the Poincaré-irreducible tensor operators of the second class is constructed. No such algebra exists for the Poincaré-irreducible tensor operators of the first class. The operator product of these tensors can only be represented as a linear superposition of the first class tensors in the limit of vanishing 4-momentum. In this limit, however, the first class tensors are no longer irreducible.