# Significance of imaginary components in Dirac probability current?

[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?

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.
• Doesn't the factor of $i$ in the gamma components and the $i$ in the current simply change the sign of the resulting solutions as they are multiplied together in the end, no? Oct 14, 2022 at 12:54
• No -- because there is a factor of $i$ between the space and time components. Dropping the overallk factor of $i$ in the current would make the $0$ component real, but then the space componts would be complex. You need to drop all the explicit "i"'s to get the usual "west coast" Bjorken and Drell metric. Oct 14, 2022 at 12:57