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 $D_{\mathit {j_{1}}}\times D_{\mathit {j_{2}}}$.These matrices have characters which are just products of the elementary characters.

$D_{\mathit {j_{1}}}\times D_{\mathit {j_{2}}}:\;\;\;\chi _{}{\mathit {j_{1}\times {\mathit {j_{2}}}}}(\phi )=\chi _{\mathit {j_{1}}}(\phi )\chi _{\mathit {j_{2}}}(\phi )$.

This expression can then be manipulated into a sum of the irreducible representation characters: $\chi _{j_{1}}(\phi )\chi _{j_{2}}(\phi )=(\sum _{\mathit {m_{2}=-j_{2}}}^{\mathit {j_{2}}}\;\;\epsilon ^{\mathit {m_{2}}})\;{\frac {\epsilon ^{\mathit {j_{1}+1}}-\epsilon ^{-{\mathit {j_{1}}}}}{\epsilon -1}}$

$=\sum _{\mathit {l=\left|j_{1}-j_{2}\right|}}^{\mathit {j_{1}+j_{2}}}\;\;{\frac {\epsilon ^{\mathit {l+1}}-\epsilon ^{\mathit {-l}}}{\epsilon -1}}$

Failed to parse (syntax error): {\displaystyle = \chi _{\mathit{j_{1}+j_{2}}}(\phi)\; +...\: +\:\chi _{\left |\mathit{j_{1}-j_{2}}\right |}(\phi)}

This shows that the product representation is reducible to a sum of the known irreducible representations:

$D_{\mathit {j_{1}}}\times D_{_{\mathit {j_{2}}}}\;=\;D_{\mathit {j_{1}+j_{2}}}+\;...\;+D_{\mathit {j_{1}-j_{2}}}.$

This is another way of aproach to the essential content of the angular momentum addition theorem.