Let
$$
\begin{gathered}
\gamma_0 = \begin{bmatrix}
0 & - i I_2 \
i I_2 & 0
\end{bmatrix} = -i \beta \
\gamma_1 = \begin{bmatrix}
-i\sigma_3 & 0 \
0 & i\sigma_3
\end{bmatrix} = - i \alpha_1 \beta \
\gamma_2 = \begin{bmatrix}
i\sigma_1 & 0 \
0 & -i\sigma_1
\end{bmatrix} = - i \alpha_2 \beta\
\gamma_3 = \begin{bmatrix}
0 & i I_2 \
-i I_2 & 0
\end{bmatrix} = - i \alpha_3 \beta \
\sigma_1 = \begin{bmatrix}
0 & 1 \
1 & 0
\end{bmatrix} \
\sigma_3 = \begin{bmatrix}
1 & 0 \
0 & -1
\end{bmatrix} \
\end{gathered}
$$
then we have $\gamma_0^2 = +1 = -\gamma_k^2$, $\gamma^k = -\gamma_k$, and $\gamma_\mu \gamma_\nu = - \gamma_\nu \gamma_\mu$ for $\mu \not= \nu$. Note that the absence of $\sigma_2$ is because Dirac desired strictly real matrices. His positive-energy relativistic wave equation is then given by
$$
\begin{gathered}
i \gamma_\mu\partial^\mu \hat q \psi = m\hat q \psi \
\iff \
(\partial^t + \alpha_k \partial^k + m\beta )\hat q \psi = 0
\end{gathered}
$$
due to $\beta^2 = (i\gamma_0)^2 = -1$, where $\hat q = \begin{bmatrix} \hat q_1 \ \hat q_2 \ -i\partial_{q_1} \ -i \partial_{q_2} \end{bmatrix}$ is a "vector operator" acting on $\psi \in \mathbb C$.
The complex wavefunction $\psi$ is designed to satisfy the Klein-Gordon $\partial^2 \psi = -m^2\psi$.
Momentum eigenstates with $i\partial \mapsto p = (p_0 + \vec p)\gamma_0$ have solutions of the form.
$$
\psi = e^{-(q_1^2 + q_2^2)/(p_0+p_3)}e^{-ip_1(q_1^2-q_2^2)/(p_0+p_3)}e^{ip_2q_1q_2/(p_0+p_3)}e^{i p\cdot x}
$$
In the rest frame, $p = p_0 \gamma_0$, we have
$$
\psi = e^{-(q_1^2 + q_2^2)/p_0}e^{i p_0 t}.
$$
The reason Dirac developed this equation because he was searching for a wave equation without negative energies. He chose this equation because $\psi$ is only normalizable for $p_0 = +m$. When $p_0 = -m$, then $\psi = e^{(q_1^2 + q_2^2)/m}e^{-i m t}$ cannot be normalized.
Dirac shows that solutions to this equation are massive particles with integral spin and no negative energies.
[[@diracPositiveenergyRelativisticWave1971]]
![[dirac-positive-energy.pdf]]
