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

Define the ladder operators $J_{\pm }=J_1\pm i{J}_2$. The SU(2) Lie algebra relations implyThus $d(J_{\pm })v_m$ has weight $m\pm 1$ when nonzero. Starting from a maximum-weight vector $v_I$, repeated lowering gives weightsThe norm formula derived from the Casimir element,shows that lowering stops precisely at $m=-I$. Therefore $2I$ is a nonnegative integer and the irreducible representation has dimension $2I+1$.

With normalized states and $J_{-}^{†}=J_{+}$,Put $n=I-m$. Iterating from the highest-weight state givesHencewhere the factorial arguments are integers because $I\pm m\in {\mathbb{Z}}_{\ge 0}$.

The transformed generators areFor example,The other two cyclic commutators work identically, soConjugation by $C$ is therefore an automorphism of a Lie algebra.

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.