momentum four vector and dirac matrices

Question

$$c\left(\alpha _i\right.{\cdot P + \beta mc) \psi = E \psi } $$ From the above dirac equation it can be shown for zero momenta that spin and antimatter are associated with $\beta $.

On the other hand, how are each of the momenta $P_x$,$P_y$,$P_z$ associated with $\alpha _1$,$\alpha _2$,$\alpha _3$? I mean does each $\alpha _1$,$\alpha _2$,$\alpha _3$ correspond directly with $P_x$,$P_y$,$P_z$ ?

OR

are they related to the Four-Momenta $P_4$ of Minkowski space, and can I write $P_4$ as

$$ \left( \begin{array}{c} E/c \\ p_x \\ p_y \\ p_z \end{array} \right) $$

I understand they are covariant, which follows naturally from the Dirac equation, but I just can't see how $\alpha _1$,$\alpha _2$,$\alpha _3$ operate to yield the momentum $P_x$,$P_y$,$P_z$.

Answer 1

Let's consider a massless Dirac fermion. In this case, we get rid of $\beta$, only three $\alpha_i$ to be determined. These matrices should anti-commute with one another and square to $\bf 1$. Pauli matrices will suffice and they are genuine spins! Then you have the correspondence: $$p^\mu \rightarrow \sigma^\mu$$ where $\sigma^0=\bf 1$.

Now consider a finite mass. This time, three 2x2 Pauli matrices will not work since we need four $(\alpha;\beta)$. The solution is enlarging the matrix dimension to 4x4 by taking tensor product $\Gamma=\sigma\otimes\tau$. We have to introduce a pseudo-spin $\tau$, which corresponds to particle-hole index. You can still make some sort of indentification of spin and momentum by looking at the $\sigma$ part.

Answer 2

$\alpha_1$,$\alpha_2$,$\alpha_3$ equal operators of components of velocity up to a constant factor (see Dirac, Principles of quantum mechanics, or just calculate commutators of coordinate operators with the Hamiltonian).