The following unnormalized vectors are solutions to the Dirac equation.
\begin{align*} u_1&=\begin{pmatrix}E+m\\0\\p_z\\p_x+ip_y\end{pmatrix} \exp\left(\frac{i\phi}{\hbar}\right) % & v_1&=\begin{pmatrix}E-m\\0\\p_z\\p_x+ip_y\end{pmatrix} \exp\left(-\frac{i\phi}{\hbar}\right) \\[1ex] u_2&=\begin{pmatrix}0\\E+m\\p_x-ip_y\\-p_z\end{pmatrix} \exp\left(\frac{i\phi}{\hbar}\right) % & v_2&=\begin{pmatrix}0\\E-m\\p_x-ip_y\\-p_z\end{pmatrix} \exp\left(-\frac{i\phi}{\hbar}\right) \\[1ex] u_3&=\begin{pmatrix}p_z\\p_x+ip_y\\E-m\\0\end{pmatrix} \exp\left(\frac{i\phi}{\hbar}\right) % & v_3&=\begin{pmatrix}p_z\\p_x+ip_y\\E+m\\0\end{pmatrix} \exp\left(-\frac{i\phi}{\hbar}\right) \\[1ex] u_4&=\begin{pmatrix}p_x-ip_y\\-p_z\\0\\E-m\end{pmatrix} \exp\left(\frac{i\phi}{\hbar}\right) % & v_4&=\begin{pmatrix}p_x-ip_y\\-p_z\\0\\E+m\end{pmatrix} \exp\left(-\frac{i\phi}{\hbar}\right) \end{align*}
where
\begin{equation*} E=\sqrt{p_x^2+p_y^2+p_z^2+m^2} \end{equation*}
and
\begin{equation*} \phi=p_xx+p_yy+p_zz-Et \end{equation*}
Of course, quantum electrodynamics uses the following solutions (in their normalized form) because the mass term $m$ is positive.
\begin{align*} u_1 &\quad \text{fermion, spin up} \\ u_2 &\quad \text{fermion, spin down} \\ v_3 &\quad \text{anti-fermion, spin up} \\ v_4 &\quad \text{anti-fermion, spin down} \end{align*}
What other combinations of $u$ and $v$ could be used as a basis for quantum electrodynamics? Are there any theories that use a different basis?
There are ${8\choose4}=70$ ways to choose four vectors, but not all of them are linearly independent.
