Dirac matrices

Four Hermitian matrices, denoted by ${\alpha }_1$, ${\alpha }_2$, ${\alpha }_3$ and $\beta$, of dimension $4\times 4$ that satisfy the relations $$\begin{array}{c}{\alpha }_k{\alpha }_j+{\alpha }_j{\alpha }_k=2{\delta }_{kj}{\mathsf{I}}_4,\\ {\alpha }_k\beta +\beta {\alpha }_k={\mathbf{0}}_4,\\ {\alpha }_k{\alpha }_k={\beta }^2={\mathsf{I}}_4,\end{array}$$ 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 $${\gamma }^{\alpha }{\gamma }^{\beta }+{\gamma }^{\beta }{\gamma }^{\alpha }=-2{\eta }^{\alpha \beta }{\mathsf{I}}_4,\mathrm{\forall }\alpha ,\beta \in \{0,1,2,3\}.$$ Here, ${\eta }^{\alpha \beta }\stackrel{\text{df}}{=}\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& -1& 0& 0\\ 0& 0& -1& 0\\ 0& 0& 0& -1\end{array}\right]$. 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 $${\gamma }^0=-i\left[\begin{array}{cc}{\mathsf{I}}_2& {\mathbf{0}}_2\\ {\mathbf{0}}_2& -{\mathsf{I}}_2\end{array}\right];{\gamma }^k=-i\left[\begin{array}{cc}{\mathbf{0}}_2& {\mathit{\sigma }}_k\\ -{\mathit{\sigma }}_k& {\mathbf{0}}_2\end{array}\right],$$ where the ${\mathit{\sigma }}_k$’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: $$(◻-m^2)E\psi =(\sum _{k=0}^3{\gamma }^k\frac{\partial }{\partial x^k}-mE)(\sum _{l=0}^3{\gamma }^l\frac{\partial }{\partial x^l}+mE)\psi =0,$$ where $◻$ denotes the d’Alembert operator.

The Dirac matrices were Introduced by P. Dirac in 1928, in his derivation of the Dirac equation.

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For references, see the article on the Dirac equation.