What does the wavefunction column mean in the Dirac equation?

Question

I'm having trouble understanding what does the column wavefunction - with 4 components - in the Dirac equation really mean (physically). How does an operator, say, the momentum operator, act on the wavefunction (does it act with the grad on each component)? And more generally what do the 4 different components represent?

Answer 1

How does an operator, say, the momentum operator, act on the wavefunction (does it act with the grad on each component)?

Yes, the momentum operator (let us say its $x$-component $p_x=-i\hbar \partial_x$) acts on the spinor wave function $\psi$ equally on each component: $$p_x \begin{pmatrix} \psi_1 \\ \psi_2 \\ \psi_3 \\ \psi_4 \end{pmatrix} = -i\hbar \begin{pmatrix} \partial_x\psi_1 \\ \partial_x\psi_2 \\ \partial_x\psi_3 \\ \partial_x\psi_4 \end{pmatrix}$$ And other operators, for example position ($x$) or orbital angular momentum ($L_x$), behave in a similar way.

However there are also operators acting differently on the spinor components, for example the spin angular momentum. Its $z$-component $S_z$ is a $4\times 4$-matrix, operating on the spinor wave function like this: $$S_z \begin{pmatrix} \psi_1 \\ \psi_2 \\ \psi_3 \\ \psi_4 \end{pmatrix} = \frac{1}{2}\hbar \begin{pmatrix} 1&0&0&0 \\ 0&-1&0&0 \\ 0&0&1&0 \\ 0&0&0&-1 \end{pmatrix} \begin{pmatrix} \psi_1 \\ \psi_2 \\ \psi_3 \\ \psi_4 \end{pmatrix} = \frac{1}{2}\hbar \begin{pmatrix} \psi_1 \\ -\psi_2 \\ \psi_3 \\ -\psi_4 \end{pmatrix}$$

And more generally what do the 4 different components represent?

The $\gamma$ matrices used in Dirac's equation are not unique. There are several representations in use. And hence the meaning of the $4$ components of the spinor wave function $\psi$ depends on which $\gamma$ representation we use. For concreteness here I will presume the so-called Dirac representation of the $\gamma$ matrices $$\gamma^0=\begin{pmatrix}1&0\\0&-1\end{pmatrix},\quad \gamma^1=\begin{pmatrix}0&\sigma_x\\-\sigma_x&0\end{pmatrix},\quad \gamma^2=\begin{pmatrix}0&\sigma_y\\-\sigma_y&0\end{pmatrix},\quad \gamma^3=\begin{pmatrix}0&\sigma_z\\-\sigma_z&0\end{pmatrix}$$ (with $0$, $1$, $\sigma_{x,y,z}$ meaning the zero, unity and Pauli $2\times 2$-matrices).
Then, in the spinor wave function of a fermion (let us say an electron) $$\begin{pmatrix} \psi_1 \\ \psi_2 \\ \psi_3 \\ \psi_4 \end{pmatrix}$$ the upper two components ($\psi_1, \psi_2$) ‒ loosely speaking ‒ represent the electron part, and the lower two components ($\psi_3, \psi_4$) represent the positron part. And within each of these two pairs of components the upper component ($\psi_1, \psi_3$) represents the spin-up part, and the lower component ($\psi_2, \psi_4$) represents the spin-down part of the electron or positron, respectively.