Sakurai's QM 8.3.1, angular momentum for the Dirac's equation

Question

In the 3rd edition of Sakurai's QM section 8.3 on symmetries of the Dirac equation, he writes (page 497, equation 8.79, he writes

$$[\boldsymbol{\alpha\cdot p},L_i]=-i\epsilon_{ijk}\alpha_jp_k\not=0.$$

Here $\boldsymbol{\alpha}$ are the three $4\times4$ matrices in the Dirac's equation and $p$ is the momentum and $L_i$ is the $i$-th component of the angular momentum. I am confused about two things:

1. How do $x_i,p_i,L_i$ etc act on the spinor $\Psi$? My understanding is that $\Psi$ is a vector with 4 components, each component is a function defined on $\mathbb{R}\times\mathbb{R}^3$, so do they just act component wise?
2. How does the commutator $[\boldsymbol{\alpha\cdot p},L_i]$ makes sense? $\boldsymbol{\alpha\cdot p}$ seems to be a four by four matrix of operators but $L_i$ is just a single operator.

Please help me with my confusion, thank you!

Answer 1

Angular momentum is diagonal in Dirac indices so you have $ [\boldsymbol{\alpha} \cdot \boldsymbol{p} , L_j]=[\alpha_i p_i, L_j]=\alpha_i[p_i,L_j] $