In Special Relativity we have the equation
$$ E^2-p_x^2-p_y^2-p_z^2=m^2c^4, $$which reflect the length of the four-momentum vector. If you try to quantize this equation you arrive to the Klein-Gordon equation.
Dirac idea: to have an equation first-order in time, analogous to Schrodinger equation, Dirac had to use matrices instead of scalars
$$ \alpha_x = \begin{pmatrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{pmatrix}, \quad \alpha_y = \begin{pmatrix} 0 & 0 & 0 & -i \\ 0 & 0 & i & 0 \\ 0 & -i & 0 & 0 \\ i & 0 & 0 & 0 \end{pmatrix}, \quad \alpha_z = \begin{pmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \\ 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \end{pmatrix}, \quad \beta = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{pmatrix} $$ $$ E = \alpha_x p_x + \alpha_y p_y + \alpha_z p_z + \beta m $$ $$ i \gamma^\mu \partial_\mu \psi - m\psi = 0 $$The solutions are not complex-valued functions, but spinor-valued functions.
If we take the non-relativistic limit, the result is not Schrodinger equation, but Pauli equation.
________________________________________
________________________________________
________________________________________
Author of the notes: Antonio J. Pan-Collantes
INDEX: