How Do Different Dirac Matrix Choices Impact RQM Equations?

[Hymne]

Hello!
I'm trying to write an essay on RQM. The problem I have encountered is the diffrent choices of matrices for the dirac equation.

The two choices that I´m mixing up in my equations are:

\begin{eqnarray}
\gamma^0 = \left( \begin{array}{cc}
I & 0 \\
0 & -I \end{array} \right), \quad
&&\gamma^1 = \left( \begin{array}{cc}
0 & \sigma_1 \\
-\sigma_1 & 0 \end{array} \right), \quad
\gamma^2 = \left( \begin{array}{cc}
0 & \sigma_2 \\
-\sigma_2 & 0 \end{array} \right) \\, \quad
&&\gamma^3 = \left( \begin{array}{cc}
0 & \sigma_3 \\
-\sigma_3 & 0 \end{array} \right),
\end{eqnarray}

And

\begin{eqnarray}
\boldsymbol{\alpha} = \left( \begin{array}{cc}
\boldsymbol{0} & \boldsymbol{\sigma_i} \\
\boldsymbol{\sigma_i} & \boldsymbol{0} \end{array} \right), \quad
&&\beta = \left( \begin{array}{cc}
\boldsymbol{1} & \boldsymbol{0} \\
\boldsymbol{0} & -\boldsymbol{1} \end{array} \right),
\label{dirachpaulimatris}
\end{eqnarray}

I clearly need to work with just one of them.
What are the benefits of working with the former respectivly the latter representation? :/

 

[Bill_K]

There's no reason not to use both. The γ's are useful for relativistic problems, while α, β are useful in the nonrelativistic limit, the relationship being γ⁰ = β, γⁱ = β αⁱ