Past exam of the particle physics course of the University of Oxford / 2016 / 3 / 2 / Solution

Embed the two off-diagonal Pauli matrices in each of the $(u,d)$, $(u,s)$ and $(d,s)$ two-dimensional subspaces. Together with two independent embedded diagonal Pauli matrices, this produces the Gell-Mann matrices:To construct the last matrix explicitly, the $(u,s)$ diagonal embedding is $D_{us}=\mathrm{diag}(1,0,-1)$ and the $(d,s)$ embedding is $D_{ds}=\mathrm{diag}(0,1,-1)$. Their difference is ${\lambda }_3$, and their sum divided by $\sqrt{3}$ is ${\lambda }_8$. There are only two independent diagonal generators.

Each matrix is Hermitian and traceless. Their inner products satisfy $\mathrm{tr}({\lambda }_a{\lambda }_b)=2{\delta }_{ab}$, proving linear independence. With $T_a={\lambda }_a/2$, arbitrary transformations near the identity have $U=\exp (i{\theta }_a{T}_a)$. The eight matrices span the generators; they are not a vector-space basis of group elements themselves.