Continuous symmetries are represented by unitary operators $U(\alpha)=e^{-i\alpha G/\hbar}$, where the Hermitian generator $G$ is conserved when $[G,H]=0$. Spatial translations, rotations, and time translations are the canonical examples.

The components satisfy $[J_i,J_j]=i\hbar\,\epsilon_{ijk}J_k$. The ladder operators $J_\pm=J_x\pm iJ_y$ connect states within an $SU(2)$ multiplet labeled by $j$ and $m$.

For two subsystems, the total space is $\mathcal{H}_1\otimes\mathcal{H}_2$. Clebsch–Gordan coefficients implement the decomposition $j_1\otimes j_2=\bigoplus_j j$, enabling spin coupling, spectroscopy, and tensor operator analysis.

Matrix elements of spherical tensor operators obey the Wigner–Eckart theorem, which separates geometric dependence from reduced matrix elements. This greatly simplifies transitions between angular-momentum eigenstates.

Spin-$\tfrac12$ states realize projective representations of $SO(3)$ and genuine representations of its double cover $SU(2)$. This is why a $2\pi$ rotation can change a spinor's sign even though measurable probabilities remain invariant.