Total angular momentum operators

[ShayanJ]

Sometimes the concept of angular momentum is presented using the idea of total angular momentum J. In those cases, its always said that we have \vec{J}=\vec L + \vec S. But I can't understand how that's possible. Because orbital angular momentum operators are differential operators and so are infinite dimensional matrices while spin angular momentum operators are finite dimensional matrices.But matrices of different dimensions can't be added. So how is that possible? What is \vec J like?
Thanks

 

[rubi]

The Hilbert space of a particle with spin $s$ is really $L^2(\mathbb R^3)\otimes \mathbb C^{2s+1}$ and $\vec{J} = \vec{L} + \vec{S}$ is just an abbreviation for $J_i = L_i\otimes\mathrm{id} + \mathrm{id}\otimes S_i$, where $L_i = \epsilon_{ijk} x_j \partial_k$ and $S_i=\frac{\hbar}{2}\pi_s(\sigma_i)$, where $\pi_s$ is a representation of $\mathfrak{sl}(2,\mathbb C)$ of dimension $2s+1$. In case of spin $s=\frac{1}{2}$, $S_i = \frac{\hbar}{2}\sigma_i$ and $\sigma_i$ are the Pauli matrices for example.