What are some examples of multiplicative quantum numbers?

Question

Wikipedia does not have much on its page for multiplicative quantum numbers, so I was wondering if there was a list or something somewhere?

Answer 1

Multiplicative quantum numbers arise as eigenvalues of finite transformations, where additive ones occurs as a result of infinitesimal transformations.

Thus, parity is multiplicative since parity is a finite transformations. Similarly, if a state $\vert\psi\rangle$ is even under permutation and $\vert\chi\rangle$ is odd under permutation of its constituents, then $\vert\psi\rangle\vert\chi\rangle$ is odd.

On the other hand, the eigenvalues of $\hat L_z$ are additive since $\hat L_z$ is a generator of infinitesimal transformations. Thus, for small $\theta$: \begin{align} \exp^{-i\theta (\hat L_z^{(1)}+\hat L_z^{(2)})}\vert \ell_1m_1\rangle \vert \ell_2m_2\rangle&\approx \left(\hat 1 -i\theta (\hat L_z^{(1)}+\hat L_z^{(2)})+\ldots\right)\vert \ell_1m_1\rangle \vert \ell_2m_2\rangle\, , \\ &= \left(\hat 1 -i\theta (m_1+m_2)\right)\vert \ell_1m_1\rangle \vert \ell_2m_2\rangle \end{align}