# How do charge conjugate fields transform under $SU(2)$ and $SU(3)$?

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.

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

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.