Phy5646/character

Angular Momentum Addition by Characters

Rotation matrices $e^{{\mathit {i}}{{\vec {\omega }}.{\vec {J}}}}$ are $(2{\mathit {j}}+1)\times (2{\mathit {j}}+1)$ matrix functions of rotating angles ${\vec {\omega }}$ in some representation of spin ${\mathit {j}}$. To indicate more explicitly the representation we are in we write them as $D_{\mathit {j}}({\vec {\omega }})$.Let us define the character by $\chi _{\mathit {j}}({\vec {\omega }})={\mathit {tr}}\;D_{\mathit {j}}({\vec {\omega }})$ For a rotation about the z-axis, the rotation matrix is diagonal

$D_{\mathit {j}}(\phi {\hat {z}})=\;diag(e^{{\mathit {ij}}\phi }e^{{\mathit {i(j-1)}}\phi }...e^{-{\mathit {ij}}\phi })$

and the character is easy to compute $\chi _{\mathit {j}}(\phi )=\sum _{\mathit {m=-j}}^{\mathit {j}}\;e^{\mathit {im\phi }}$ $={\frac {\epsilon ^{\mathit {j+1}}-\epsilon ^{-{\mathit {j}}}}{\epsilon -1}}\;\;\;where\;\;\epsilon =e^{\mathit {i\phi }}$ $={\frac {sin({\mathit {j+{\frac {1}{2}})\phi }}}{sin({\frac {\phi }{2}})}}$

But any rotation may be brought to diagonal form by a similarity transform, so this is the most general character. It depends on the rotation angle, not the direction. If we tensor together the states $|{\mathit {j_{1}m_{1}}}\rangle$ and $|{\mathit {j_{2}m_{2}}}\rangle$, they transform under the tensor product representation $D_{\mathit {j_{1}}}\times D_{\mathit {j_{2}}}$.