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Choice of dirac matrices

  1. Oct 6, 2011 #1
    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:

    \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),


    \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),

    I clearly need to work with just one of them.
    What are the benefits of working with the former respectivly the latter representation? :/
  2. jcsd
  3. Oct 6, 2011 #2


    User Avatar
    Science Advisor

    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
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