Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 302 / 2 / d / Solution

Since $C{J}_3{C}^{-1}=-J_3$, the operator $C$ sends a weight-$m$ state to a weight-$-m$ state. AlsoFor the spin-one normalization,Write $C|1,1⟩=a|1,-1⟩$. Using $C|1,0⟩=|1,0⟩$ in the first relation gives $a=-1$, and applying $C$ to the second gives the other phase. Therefore, in the stated phase convention,A simultaneous phase redefinition of the charged pion states changes both displayed signs but not their physical interchange under charge conjugation.