Abstract
About 60 years ago, I. Shmushkevich presented a simple ingenious method for computing the relative probabilities of channels involving the same interacting multiplets of particles, without the need to compute the Clebsch-Gordan coefficients. The basic idea of Shmushkevich is “isotopic non-polarization” of the states before the interaction and after it. Hence his underlying Lie group was <em>SU</em>(2). We extend this idea to any simple Lie group. This paper determines the relative probabilities of various channels of scattering and decay processes following from the invariance of the interactions with respect to a compact simple a Lie group. Aiming at the probabilities rather than at the Clebsch-Gordan coefficients makes the task easier, and simultaneous consideration of all possible channels for given multiplets involved in the process, makes the task possible. The probability of states with multiplicities greater than 1 is averaged over. Examples with symmetry groups <em>O</em>(5), <em>F</em>(4), and <em>E</em>(8) are shown.
Highlights
The method of Shmushkevich [1, 2] was conceived as a simpler alternative to computing the relative probabilities of various channels of scattering and decay processes under strict isospin invariance (SU (2) invariance)
Multiplicity Orbit size we provide an illustration of this approach, for the symmetry groups O(5), F (4), and E(8)
Our aim is to show how to average over different particles/states which carry the same Lie group representation labels
Summary
The method of Shmushkevich [1, 2] was conceived as a simpler alternative to computing the relative probabilities of various channels of scattering and decay processes under strict isospin invariance (SU (2) invariance). The difficulty of generalizing of Shmushkevich’s method to higher rank groups lies in the frequent occurrence of multiple states with the same quantum numbers, equivalently labeled by the same weights of irreducible representations [9], as well as the sheer number of channels that need to be written down. It is likely that practical exploitation of Shmushkevich’s idea for higher groups and possibly representations of much higher dimensions, will not proceed by spelling out the large number of channels for each case and counting the number of occurrences of each state in all the channels. This is a routine operation which, does not produce all possible channels This will relate only the states which are situated on the same Weyl group orbit.
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