Understanding Color Factors in Eq. 8.45-8.46

[spookyfish]

Hi. I was reading Griffiths page 289 and I wasn't sure how the transition from Eq. (8.45) to (8.46) was allowed. What bothers me is factoring out the "color factors" since the c's are vectors, and generally the elements of these color factors are matrices. For example, how is it allowed to pull out $c_3^\dagger \lambda^\alpha c_1$ out of $\bar{u}(3)c_3^\dagger \lambda^\alpha \gamma^\mu u(1) c_1$. Is it because the color factors live in a different space (color space) so they commute with spinor-space matrices?

 

[Einj]

Yes, the color vectors $c_i$ live in a space different from that of the Dirac spinors $u(i)$. If you still feel uncomfortable you can always use explicit indices: the Dirac spinors and the Dirac matrices have indices, say, i,j,k and the color vectors and the color generators have indices, say, a,b,c. Once the indices are explicit all the quantities commute with each other since they are just numbers. Then, for example:
$$
\bar u(3)c_3^\dagger\lambda^\alpha\gamma_\mu u(1)c_1=\bar u_i(3)c_{3a}^\dagger\lambda^\alpha_{ab}\gamma_{\mu}^{ij}u_j(1)c_{1b}= \bar u_i(3)\gamma_\mu^{ij}u_j(1)c_{3a}^\dagger\lambda_{ab}^\alpha c_{1b}=(\bar u(3)\gamma_\mu u(1))(c_3^\dagger\lambda^\alpha c_1).
$$
I hope this answer your question.

 

[spookyfish]

yes, thank you