# Generalization of Dirac matrices

Is it possible to have a set of 16-dimensional matrices $$\gamma_{\mu}^{a}$$ such that $$\{\gamma_{\mu}^{a},\gamma_{\nu}^{b}\} = 2\delta^{ab}\eta_{\mu\nu}$$ where $$\eta_{\mu\nu}$$ is the Minkowski metric and $$a$$ and $$b$$ can be 1 or 2? I've been trying to construct them manually but it's tougher than I thought and I can't find any mention of this in the literature.

Edit: And also, such that $$\gamma^a_{\mu}\gamma^b_{\nu}$$ is not 0.

• ? It is not evident by inspection? a 16x16 matrix with the $\gamma_\mu^1$s in the upper left 4x4 block and zero in the lower right block, and $\gamma_\mu^2$s in the lower right and zero in the upper left does not qualify? $\gamma_\mu\oplus 0$ and $0\oplus \gamma_\mu$. Isn't this your algebra? – Cosmas Zachos Mar 9 at 16:44
• In that case I will also have $\gamma^{a}_{\mu}\gamma^{b}_{\nu}=0$, which I do not wish. I should've added that, thanks. – user38680 Mar 9 at 16:50
• isnt this just $C\ell(2)\otimes C\ell(1,d-1)$? if so, you know the irreps, so take as many direct sums as necessary. – AccidentalFourierTransform Mar 9 at 23:41
• Thanks for your comment @AccidentalFourierTransform. I understand what you said and it makes sense, but I don't have enough practice in group theory to actually do what you suggested. Could you quickly sketch how one would do that? Thanks! – user38680 Mar 10 at 11:58
• @ AccidentalFourierTransform . Alas, the $\sigma_1$ and $\sigma_2$ in different parts of the anticommutator flip the sign of the conventional Cl. If only one considered a commutator, instead... – Cosmas Zachos Mar 10 at 15:53