Dimension of gamma matrices in higher dimensional Dirac equations

Question

Reading about Dirac's equation in higher dimensional space-times I have read that the gamma matrices are $2^{[D/2]}\times{}2^{[D/2]}$.

So, if we have $D=11$, for example, how is this formula supposed to be understood?

Answer 1

You should understand it as rounded down. For example, in $D = 3$, you should have $2^1\times 2^1$ matrices. Indeed in $D = 3$ the Dirac matrices are the familiar Pauli matrices. $$ \sigma_1 = \frac{1}{2}\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \quad \sigma_2 = \frac{1}{2}\begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} \quad \sigma_3 = \frac{1}{2}\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$$ (you can verify that they satisfy the anticommutation relation required with respect to the Euclidean metric).