My question concerns the plane wave solutions to the Majorana equation. First, recall the Dirac equation:
$$(i\gamma^\mu \partial_\mu-m)\psi=0$$ I suggest a solution in the form of a plane wave with $\vec{p}=0$ and $c=\hbar=1$: $$\psi=\psi_0 e^{\pm imt}$$ Plugging this into the Dirac equation, I find the equation
$$\gamma^0 \psi_0=\pm \psi_0$$
This leads to four real-valued terms for $\psi_0$, which correspond to the two spin solutions for positive and negative energies. I tried doing the same exact thing for the Majorana equation, but now I have
$$\widetilde{\gamma}^0\psi_0=\pm \psi_0$$
where $\widetilde{\gamma}^0$ is given in the Majorana basis as
$$\widetilde{\gamma}^0=\begin{pmatrix} 0 & 0 & 0 & -i\\ 0 & 0 & -i & 0 \\ 0 & i & 0 & 0\\ i & 0 & 0 & 0 \end{pmatrix}$$ When I solve the above for $\psi_0$, I then get a complex solution, which doesn't make sense. I thought that, under the Majorana basis, solutions would be of the form $\psi=\psi^*$? Also, a plane wave solution should still work for the Majorana equation, because, like its Dirac counterpart, it is still a factorization of the Klein-Gordon equation. Any explanation of what I'm doing wrong would be greatly appreciated.
