Consider two spin systems $\mathcal{H}_1$ and $\mathcal{H}_2$, each with spin operator $L_1$ and $L_2$, respectively.
What is the difference between $L_1+L_2$ and $L_1 \cdot L_2$?
As far as I understand, both act component-wise on the tensor product space $\mathcal{H}_1\otimes\mathcal{H}_2$ so shouldn't they really be the same?
