Spin Conventions

The following set of $\gamma$ -matrices are used in four dimensions:

$\‍[\begin{array}{lccc} \gamma_0\quad\quad& \left(\begin{array}{rrrr} 0&0&0&i\\ 0&0&i&0\\ 0&-i&0&0\\ -i&0&0&0\\ \end{array}\right)\quad & \left(\begin{array}{rr} 0&i\sigma^1\\ -i\sigma^1&0 \end{array}\right)\quad\quad & -\sigma^2\!\otimes\!\sigma^1\\ \gamma_1\quad\quad& \left(\begin{array}{rrrr} 0&0&0&-1\\ 0&0&1&0\\ 0&1&0&0\\ -1&0&0&0\\ \end{array}\right)\quad & \left(\begin{array}{rr} 0&-i\sigma^2\\ i\sigma^2&0 \end{array}\right)\quad\quad & \sigma^2\!\otimes\!\sigma^2\\ \gamma_2\quad\quad& \left(\begin{array}{rrrr} 0&0&i&0\\ 0&0&0&-i\\ -i&0&0&0\\ 0&i&0&0\\ \end{array}\right)\quad & \left(\begin{array}{rr} 0&i\sigma^3\\ -i\sigma^3&0 \end{array}\right)\quad\quad & -\sigma^2\!\otimes\!\sigma^3\\ \gamma_3\quad\quad& \left(\begin{array}{rrrr} 0&0&1&0\\ 0&0&0&1\\ 1&0&0&0\\ 0&1&0&0\\ \end{array}\right)\quad & \left(\begin{array}{rr} 0&\mathbf{1}\\ \mathbf{1}&0 \end{array}\right)\quad\quad & \sigma^1\!\otimes\!1\\ \end{array} \‍]$

The basis is chiral. All the possible gamma matrix products are represented via

$\‍[\Gamma(n) = \gamma_0^{n_0} \gamma_1^{n_1} \gamma_2^{n_2} \gamma_3^{n_3} \‍]$

where n_i are single bit fields. Since $\gamma_0$ comes first the bit for it must come first.

So, $\gamma_5 = \gamma_0\gamma_1\gamma_2\gamma_3$ is represented as 1111b = 15d, and $\gamma_0\gamma_1\gamma_3$ is represented as 1011b = 11d (note the ordering). The conventional $\gamma$ -matrices are

$\‍[\Gamma(1) = \gamma_0\qquad \Gamma(2) = \gamma_1\qquad \Gamma(4) = \gamma_2\qquad \Gamma(8) = \gamma_3 \‍]$

This enumeration is $\gamma$ -basis independent.