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I am trying to derive the gauge transformation for the charge conjugate field of a quark doublet (left handed quark) such that its field $Q$ transforms under $SU(2)$ and $SU(3)$ as:

$SU(2):$ $Q \rightarrow \exp\left[ \frac{i}{2}\theta^{a} \sigma^{a}\right] Q$, where $\sigma^{a}$ are the Pauli matrices and $\theta^{a}$ some group parameter.

$SU(3):$ $ \rightarrow \exp\left[ \frac{i}{2}\alpha^{a} t^{a}\right] Q$, where $\alpha^{a}$ is some group parameter and the $t^{a}$ are $SU(3)$ generators in the fundamental representation, for instance Gell-Mann matrices.

Now I need to find the corresponding transformations for $SU(2)$ and $SU(3)$ for the charge conjugate field of $Q$ defined as: $Q^{c}\equiv i\gamma^{2}\gamma^{0}\bar{Q}^{T}$.

What I started to do is try to find $\delta Q^{c}$ for both cases:

$SU(2):$ $Q^{c}\rightarrow i\gamma^{2}\gamma^{0}(1-\frac{i}{2}\theta^{a}{\sigma^{a}}^{\ast}) Q^{\ast}$, then $\delta Q^{c}=i\gamma^{2}\gamma^{0}(-\frac{i}{2}\theta^{a}{\sigma^{a}}^{\ast})Q^{\ast}$ Now my question is how do I commute ${\sigma^{a}}^{\ast}$ with the gamma matrices here so I can get a $Q^{c}$ on the LHS?

I have the same question for the $SU(3)$ case:

$SU(3):$ $\delta Q^{c}=i\gamma^{2}\gamma^{0}(-\frac{i}{2}\alpha^{a}{t^{a}}^{\ast})Q^{\ast}$. Do the Gell-Mann matrices commute with the gamma matrices here or not?

Thanks a lot and sorry if this is a too much of a beginner kind of question.

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  • $\begingroup$ Are not the gamma matrices four-dimensional while the Gell- Mann matrices are three-dimensional? $\endgroup$ Mar 7, 2021 at 22:19

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The gamma matrices act on spinor indices while the $SU(2)$ and $SU(3)$ matrices act on group indices, for one is like the others do not exist. Therefore they commute.

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