Clebsch-Gordan Coefficients for the Extended Quantum-Mechanical Poincar\'e Group and Angular Correlations of Decay Products

Abstract

This paper describes Clebsch-Gordan coefficients (CGCs) for unitary irreducible representations (UIRs) of the extended quantum mechanical Poincar\'e group $\pt$. `Extended' refers to the extension of the 10 parameter Lie group that is the Poincar\'e group by the discrete symmetries $C$, $P$, and $T$; `quantum mechanical' refers to the fact that we consider projective representations of the group. The particular set of CGCs presented here are applicable to the problem of the reduction of the direct product of two massive, unitary irreducible representations (UIRs) of $\pt$ with positive energy to irreducible components. Of the sixteen inequivalent representations of the discrete symmetries, the two standard representations with $U_C U_P = \pm 1$ are considered. Also included in the analysis are additive internal quantum numbers specifying the superselection sector. As an example, these CGCs are applied to the decay process of the $\Upsilon(4S)$ meson.