quantum mechanics - Spin commutation relations

For orbital angular momentum defined as $L= r \times p $ we can prove, in quantum mechanics, the commutation relations. Also, we could prove these relationships through the study of rotations (infinitesimal) in space. These are: $$[L_i , L_j]=i \hbar \sum_k ε_{ijk}L_k. $$

Since there isn't an analogous definition for spin angular momentum like that of the orbital angular momentum,

1. 
   

   How can we prove the commutation relations: $$[S_i , S_j]= i \hbar \sum_k ε_{ijk}S_k. $$

   
   



2. 
   

   Can we follow a path similar to that of the orbital angular momentum, that is the study of rotations in some space and if yes, in what space and what would this space represent?

   
   

Answer

You appear confused by how spin is introduced in ordinary QM. It is rather ad hoc:

Given a Hilbert space without spin degrees of freedom of a particle $\mathcal{H}_0$, and the spin $s$ of the particle, we take the total space of states of the particle to be $\mathcal{H}_0\otimes \mathcal{S}_s$, where $\mathcal{S}_s$ is a $2s+1$-dimensional complex Hilbert space carrying the unique irreducible representation of $\mathrm{SU}(2)$ labeled by $s$.

By construction, there are three anti-Hermitian generators $T_i\in\mathfrak{su}(2)\cong\mathfrak{so}(3)$ acting on $\mathcal{S}_s$ fulfilling the commutation relations $$ [T_i,T_j] = \sum_k\epsilon_{ijk}T_k$$ from which you get the usual Hermitian spin operators by multiplying by $\mathrm{i}$.

For $s=1$, the space $\mathcal{S}_1$ is three-dimensional, and the action of the $T_i$ is just a real-valued rotation around the $i$-axis, but, in general, the representations of $\mathrm{SU}(2)$ are not rotations, although they may be, whenever the representation map $\mathrm{SU}(2)\to\mathrm{U}(2s+1)$ hits only the real orthogonal matrices $\mathrm{O}(2s+1)\subset\mathrm{U}(2s+1)$, which happens for integer $s$.