Dirac representation of the gamma matrices (${\tilde{\gamma }}^{\mu }$)

With signature $(+,-,-,-)$, one standard representation is ${\tilde{\gamma }}^0=\mathrm{diag}(I_2,-I_2)$ and ${\tilde{\gamma }}^i=\left(\begin{array}{cc}0& {\sigma }^i\\ -{\sigma }^i& 0\end{array}\right)$. Block multiplication and the Pauli matrix multiplication law give time-time anticommutator $2I_4$, mixed anticommutators zero, and spatial anticommutators $-2{\delta }^{ij}{I}_4$. Thus these gamma matrices represent the spacetime Clifford algebra.

For the Dirac representation of the gamma matrices, the unitary matrixsends ${\tilde{\gamma }}^0$ to $\left(\begin{array}{cc}0& I_2\\ I_2& 0\end{array}\right)$ while commuting with every spatial gamma matrix. This is the standard Weyl representation of the gamma matrices with spatial blocks $(0,{\sigma }^i;-{\sigma }^i,0)$. The similarity transformation preserves the Clifford algebra relations. Changing block-order or chirality conventions can change the displayed transformation, so its direction must be specified.