[Following the derivation presented here (pp. 498-504)]
Each plane wave spinor is a solution of the Dirac equation,
\begin{align}
\left(\gamma_\mu \frac{\partial}{\partial x_\mu}+m\right)\psi=0 \quad : \quad \gamma_0 = \begin{pmatrix} 1 &0 \\ 0 & -1 \end{pmatrix}, \ \ \gamma_k = \begin{pmatrix}0 & -i\sigma_k \\ i\sigma_k & 0 \end{pmatrix},
\end{align}
and takes the form
\begin{align}
\psi(x)=u(p)e^{ip\cdot x},
\end{align}
where $u(p)$ is a bispinor with components
\begin{align}
u^{(1)} = \sqrt{\frac{\text{E} + m}{2m }} \begin{pmatrix} 1\\ 0 \\ \frac{\text{p}_z}{\text{E} + m}\\ \frac{\text{p}_x+i\text{p}_y}{\text{E} + m}\end{pmatrix},& \quad \quad u^{(2)} = \sqrt{\frac{\text{E} + m}{2m }} \begin{pmatrix} 0\\ 1 \\ \frac{\text{p}_x-i\text{p}_y}{\text{E} + m}\\ \frac{-\text{p}_z}{\text{E} + m}\end{pmatrix}, \\ u^{(3)} = \sqrt{\frac{-\text{E} + m}{2m }} \begin{pmatrix} \frac{-\text{p}_z}{-\text{E} + m}\\ \frac{-(\text{p}_x+i\text{p}_y)}{-\text{E} + m} \\ 1\\ 0\end{pmatrix},& \quad \quad u^{(4)} = \sqrt{\frac{-\text{E} + m}{2m }} \begin{pmatrix} \frac{-(\text{p}_x-i\text{p}_y)}{-\text{E} + m}\\ \frac{\text{p}_z}{-\text{E} + m} \\ 0\\ 1 \end{pmatrix}.
\end{align}
The solutions can be further understood in terms of the conserved probability current,
\begin{align}
j_\mu = i \bar \psi \gamma_\mu \psi,
\end{align}
where $\bar \psi = \psi^\dagger \gamma_0$. Written in terms of $\psi = (A_1 \ A_2 \ B_1 \ B_2)^T$, we have
\begin{align}
j_\mu &= i\begin{pmatrix} A_1^* & A_2^* & -\!B_1^* & -\!B_2^*\end{pmatrix}\gamma_\mu \begin{pmatrix}
A_1 \\ A_2 \\ B_1 \\ B_2 \end{pmatrix}\\ &= \begin{pmatrix}
i[A_1^*A_1 + A_2A_2^* + B_1^*B_1 + B_2B_2^*]\\
A_1^*B_2 + A_1B_2^* + A_2^*B_1 + A_2B_1^* \\
-i[A_1^*B_2 – A_1B_2^* – A_2^*B_1 + A_2B_1^*] \\
A_1^*B_1 + A_1B_1^* – A_2^*B_2 – A_2B_2^*
\end{pmatrix}.
\end{align}
If we compare the probability currents of the particle and anti-particle plane waves, we find that
\begin{align}
\begin{cases}
j_\mu^{(1)} = j_\mu^{(2)} = \frac{1}{|E|}\begin{pmatrix} i\text{E},& \text{p}_x, & \text{p}_y, & \text{p}_z \end{pmatrix}, \\
j_\mu^{(3)} = j_\mu^{(4)} = \frac{1}{|E|}\begin{pmatrix} i\text{E},& -\text{p}_x, & -\text{p}_y, & -\text{p}_z \end{pmatrix},
\end{cases}
\end{align}
where the $1/|E|$ is a normalization factor.
Can anyone provide an explanation for, or reveal any deeper significance to, the temporal component of $j$ having an imaginary value? As this is a four-vector, the components can mix under Lorentz transformations. This appears to imply that the flow of probability is complex-valued in general. Or am I missing something?
Best Answer
All the gamma matrices you quote obey $(\gamma^\mu)^2=1$, so the author of the notes seems to be using the obsolete $x^4=ict$ convention. In this metric convention the Lorentz group is obscured by making it look like 0(4) and the time components of physical quantities are complex. A modern version would have no factor of $i$ in the current and no $i$'s multiplying the $\sigma$'s in the other gamma matrices. Whose notes are these? I'd get a better set.
Edit: I just checked, and yes the author of the notes states on page 46 that he/she uses the $(x,y,z,ict)$ convention that gives and illusion of mathematical simplicity at the expense of obscuring the physics. This convention has not been used in field theory since the 1950's.