Spin-one half-turn matrix
= Spin-one half-turn matrix
{title2=$e^{-i\pi J_2}=\begin{pmatrix}0&0&1\\0&-1&0\\1&0&0\end{pmatrix}$}
In the ordered spin-one basis $(m=1,0,-1)$, the second <angular momentum operator> satisfies $J_2^3=J_2$. Therefore its <matrix exponential> reduces to $I-i\sin\theta J_2+(\cos\theta-1)J_2^2$. A half-turn exchanges the two outer weights and negates the middle weight. Applied to the <pion> <isospin> triplet, this fixes the rotation factor in <G parity>.