Gamma matrices

Description

Let $\eta = \mathrm{diag}(1, -1, -1, -1)$ be the Minkowski metric, with indices $\mu, \nu \in \set{0, 1, 2, 3}$. Gamma matrices, or Dirac matrices, are four complex $4 \times 4$ matrices $\gamma^0, \gamma^1, \gamma^2, \gamma^3$ that satisfy $$\gamma^\mu \gamma^\nu + \gamma^\nu \gamma^\mu = 2\eta^{\mu\nu} I_4 \qquad \text{for all } \mu, \nu.$$ The standard choice, the Dirac representation, is written in $2 \times 2$ blocks as $$\gamma^0 = \begin{pmatrix} I_2 & 0 \\ 0 & -I_2 \end{pmatrix}, \qquad \gamma^k = \begin{pmatrix} 0 & \sigma_k \\ -\sigma_k & 0 \end{pmatrix} \quad (k = 1, 2, 3),$$ where $\sigma_1, \sigma_2, \sigma_3$ are the Pauli matrices.

Check: $(\gamma^0)^2 = I_4$. Since $\sigma_k^2 = I_2$, the block product gives $(\gamma^k)^2 = \mathrm{diag}(-\sigma_k^2, -\sigma_k^2) = -I_4$. The product $\gamma^0\gamma^k$ has off-diagonal blocks $\sigma_k$ and $\sigma_k$, and $\gamma^k\gamma^0$ has off-diagonal blocks $-\sigma_k$ and $-\sigma_k$, so they anticommute. For $j \ne k$, $\gamma^j\gamma^k + \gamma^k\gamma^j = -\mathrm{diag}(\sigma_j\sigma_k + \sigma_k\sigma_j, \sigma_j\sigma_k + \sigma_k\sigma_j) = 0$, since distinct Pauli matrices anticommute.