Someone argues:"The generators of rotation obey and cyclic permutations
PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Quantum Mechanics · Symmetry, angular momentum and spin
The Structure of Quantum Mechanics, Ch Angular momentum from the rotation group
Problem
Someone argues:"The generators of rotation obey [J^x,J^y]=iℏJ^z and cyclic permutations. This is exactly the algebra used for orbital angular momentum L^, and there the ladder argument showed that the top rung of a multiplet forces l to be a non-negative integer, l=0,1,2,…. Since J^ obeys the identical algebra, the identical ladder argument must force j to be a non-negative integer for any angular momentum whatsoever, spin included. So a spin quantum number of 21 is really a bookkeeping convenience for two states that ought to be labelled by an integer j in some other way."Find the error in this reasoning, stating precisely what the ladder argument from the algebra alone does establish about the allowed values of j, and what extra physical requirement is needed to cut that list down to integers only for the orbital case.