0
$\begingroup$

The generators of rotations $J_i$ under rotation transform as $$J_i'=R_{ij}J_j\,.$$

Now $J^2=J_1J_1+J_2J_2+J_3J_3$ trasform as

$$ \begin{aligned} J'^2&=RJ^2R^{-1}=RJ_1R^{-1}RJ_1R^{-1}+RJ_2R^{-1}RJ_2R^{-1}+RJ_3R^{-1}RJ_3R^{-1}\\&=R_{1j}R_{1k}J_kJ_j+R_{2j}R_{2k}J_kJ_j+R_{3j}R_{3k}J_kJ_j\,. \end{aligned}$$

Now $J^2$ is a scalar so it should not transform like this.

What am I doing wrong here?

$\endgroup$
3
$\begingroup$

$R$, being a rotation, is orthogonal $$ R_{1k}R_{1j} + R_{2k}R_{2j}+R_{3k}R_{3j} = \sum_{i=1}^3 R_{ik}R_{ij} = (R^T R)_{kj} = \delta_{ij}\,.$$ Then $$ J_k J_j \delta_{kl} = J^2\,. $$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.