My question occurred when I was reading Introduction to Elementary Particles by David J. Griffiths. In chapter 8, part 8.5, he is calculating the colour factor of quark-antiquark annihilation.
My question relies in the derivation of three formulas, from (8.82) to (8.84), which read: $$\mathcal{f}=\frac{1}{2 \sqrt{3}} a^{\alpha}_3 a^{\alpha}_4$$ $$|\text{singlet}\rangle=\frac{1}{\sqrt{8}}\sum^8_{n=1} |n_1\rangle |n_2\rangle$$ $$a^{\alpha}_3 a^{\alpha}_4= \frac{1}{8}(8)=1$$ I am puzzled by the derivation through these three formulas, does $a^{\alpha}_3$ or $a^{\alpha}_4$ exactly refer to the singlet state? If so, what does the summation over the index $\alpha$ mean? What is the relation between this summation and normalisation of the singlet state?
