Question about the Dirac equation Energy and momentum of a particle can be expressed by equation 
$$E^2=p_1^2c^2+p_2^2c^2+p_3^2c^2+m^2c^4\hspace{40pt}(1)$$
Equation (1) can be divided into $E$ on both sides. We obtain
$$E=\frac{v_1}{c}p_1\,c+\frac{v_2}{c}p_2\,c+\frac{v_3}{c}p_3\,c+\frac{v_4}{c}m\,c^2\hspace{40pt}(2)$$ where $v^2=v_1^2+v_2^2+v_3^2$, and $v_4=\sqrt{c^2-v^2}$;
The Dirac equation has the form $$i\hbar\frac{\partial\psi}{\partial t}=(\alpha_1\hat p_1c+\alpha_2\hat p_2c+\alpha_3\hat p_3c+\alpha_4m\,c^2)\psi\hspace{30pt}(3)$$ where $\alpha_i$ is matrix $(i=1,2,3,4)$.
From the principle of correspondence between (2) and (3) is $\alpha_i\rightarrow v_i/c$.
In quantum mechanics, it is shown that the relativistic velocity operator $v_v=dx_v/dt$; $(v=1,2,3)$ is given by $\hat{v}_v=c\,\alpha_v$, ie is a matrix operator. Then relativistic velocity operator $v_4=\sqrt {c^2-v^2}$ is the matrix $\hat v_4=c\alpha_4$. Is it right? Does the equation (2) be the basis of the Dirac equation?
 A: The Dirac equation (according to my understanding), is an alternate form of the Klein-Gordon equation. The Dirac equation can be derived from the following equation.
$$ Z_{μ\ }Z^{μ}-m^{2}=0 $$
 Were,$$ Z_{μ\ }= \left(E\ \ \ P\right)$$
and,
$$Z^{μ}= \begin{pmatrix}
           E\\-P \end{pmatrix} $$
Where "E" is the operator of energy, and "P" is the operator of momentum. The equation:
$$ Z_{μ\ }Z^{μ}-m^{2}=0 $$
Is the 4-vector representation of the equation:
$$ E^{2}-P^{2}-m^{2}=0 $$
Which is just a rearranged form of the Mass-Energy equation, with natural-unit simplification namely,
$$E^{2}=\ P^{2}+m^{2\ }$$
Therefore, the Dirac equation is just the Schrodinger equation, with a relativistic Hamiltonian. I guess, if you start with a relativistic Hamiltonian, and rearrange it in 4-vector notation, and then substitute in Quantum operators for energy and momentum, you will eventually get some form of the Dirac equation! 
The connection between the equation,
$$ Z_{μ\ }Z^{μ}-m^{2}=0 $$
and the Dirac equation, can be found in this video: https://www.youtube.com/watch?v=jjG2Y_dMsbI
I hope this helps!
