# Dirac matrices

Four Hermitian matrices, denoted by $\alpha_{1}$, $\alpha_{2}$, $\alpha_{3}$ and $\beta$, of dimension $4 \times 4$ that satisfy the relations \begin{gather} \alpha_{k} \alpha_{j} + \alpha_{j} \alpha_{k} = 2 \delta_{k j} \mathsf{I}_{4}, \\ \alpha_{k} \beta + \beta \alpha_{k} = \mathbf{0}_{4}, \\ \alpha_{k} \alpha_{k} = \beta^{2} = \mathsf{I}_{4}, \end{gather} where $\mathsf{I}_{4}$ is the $(4 \times 4)$ identity matrix. The matrices $\alpha_{1}$, $\alpha_{2}$, $\alpha_{3}$ and $\beta$ may also be replaced by the Hermitian matrices $\gamma^{k} = - i \beta \alpha_{k}$, where $k \in \{ 1,2,3 \}$, and by the anti-Hermitian matrix $\gamma^{0} = i \beta$. These then satisfy the relation $$\gamma^{\alpha} \gamma^{\beta} + \gamma^{\beta} \gamma^{\alpha} = - 2 \eta^{\alpha \beta} \mathsf{I}_{4}, \qquad \forall \alpha,\beta \in \{ 0,1,2,3 \}.$$ Here, $\eta^{\alpha \beta} \stackrel{\text{df}}{=} \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{bmatrix}$. It is therefore possible to write the Dirac equation in a form that is covariant with respect to the Lorentz group of transformations. The matrices $\alpha_{k}$, $\beta$ and $\gamma^{k}$, where $k \in \{ 0,1,2,3 \}$, are defined up to an arbitrary unitary transformation and may be represented in various ways. One such representation is $$\gamma^{0} = - i \begin{bmatrix} \mathsf{I}_{2} & \mathbf{0}_{2} \\ \mathbf{0}_{2} & - \mathsf{I}_{2} \end{bmatrix}; \qquad \gamma^{k} = - i \begin{bmatrix} \mathbf{0}_{2} & \boldsymbol{\sigma}_{k} \\ - \boldsymbol{\sigma}_{k} & \mathbf{0}_{2} \end{bmatrix},$$ where the $\boldsymbol{\sigma}_{k}$’s are the $(2 \times 2)$ Pauli matrices, while $\mathsf{I}_{2}$ and $\mathbf{0}_{2}$ are the $(2 \times 2)$ identity and zero matrices respectively. Dirac matrices may be used to factorize the Klein–Gordon equation in the following manner: $$(\Box - m^{2}) E \psi = \left( \sum_{k = 0}^{3} \gamma^{k} \frac{\partial}{\partial x^{k}} - m E \right) \! \left( \sum_{l = 0}^{3} \gamma^{l} \frac{\partial}{\partial x^{l}} + m E \right) \psi = 0,$$ where $\Box$ denotes the d’Alembert operator.