A confusion about $SU(2)$ group

Question

I was reading the basic properties about $SU(2)$ group. As the fundamental representation, $SU(2)$ matrices operate on two-dimensional vector $$ \eta=\begin{pmatrix}\eta_1 \\\eta_2\end{pmatrix}. $$ The general transformation matrix of $SU(2)$ can be written as $$ \hat{U}=\begin{pmatrix}a &b \\-b^* &a^*\end{pmatrix}\,,\qquad\tilde{\eta}=\hat{U}\eta. $$ I was told that $\eta$ and $\eta^\dagger$ transform in a different way under $SU(2)$. It is easy to understand since $$ \tilde{\eta}=\hat{U}\eta\,,\qquad\tilde{\eta}^\dagger=\tilde{\eta}^\dagger\hat{U}^\dagger. $$ However, it is confused to be told that $$ \begin{pmatrix}\eta_1 \\\eta_2\end{pmatrix}\,,\qquad \begin{pmatrix}-\eta^*_2 \\\eta^*_1\end{pmatrix} $$ transform in the same way. My question is what is the meaning of this sentence?

Answer 1

The sentence introduces the celebrated conjugate representation of SU(2), the very linchpin of fermion masses for both members of chiral doublets in a generation of the SM.

Defining $$ \theta\equiv -i\sigma_2 \eta^*, $$ you have $$ \eta\mapsto \hat U \eta, \implies \eta^*\mapsto \hat U^* \eta^* , \implies \\ \theta \mapsto -i\sigma_2 \hat U^* (i\sigma_2) \theta= \hat U \theta. $$

Answer 2

First, let me mention that a matrix of the form $$ U=\begin{pmatrix}a &b \\-b^* &a^*\end{pmatrix}$$

is only an element of $\operatorname{SU}(2)$ if $\det U = 1$. With that being said, let's look at your question:

Let $U=\begin{pmatrix}a &b \\-b^* &a^*\end{pmatrix} \in \operatorname{SU}(2)$. Consider first $$ \begin{pmatrix} \tilde \eta_1 \\ \tilde \eta_2\end{pmatrix} := U \cdot \begin{pmatrix} \eta_1 \\ \eta_2\end{pmatrix} = \left(\begin{matrix} a\eta_1+b\eta_2 \\ \eta_2 a^*-\eta_1 b^* \end{matrix}\right)\,. $$ Now consider $$ \begin{pmatrix} -\tilde \eta_2^* \\ \tilde \eta_1^*\end{pmatrix} := U \cdot \begin{pmatrix} -\eta_2^* \\\eta_1^*\end{pmatrix} =\begin{pmatrix} b \eta_1^* - a \eta_2^*\\ a^* \eta_1^* + b^* \eta_2^* \end{pmatrix}\,. $$ It is easy to see that the resulting transformations for $\eta_1, \eta_2, \eta_1^*, \eta_2^*$ are consistent with each other. In this sense, one can say that $\begin{pmatrix} \eta_1 \\ \eta_2 \end{pmatrix} $ and $\begin{pmatrix} -\eta_2^* \\\eta_1^*\end{pmatrix}$ transform in the same way.