Question about choosing the time reversal operator According to the "QFT Nutshell" page 102-104 about Time reversal, the author said about applying the time reversal to the spin 1/2 system. 

Let $T=UK$, where $K$ complex conjugates everything to its right and $U$ is a unitary operator. Acting with $T$ on the spin-up state, we obtain the spin-down state. Thus, we need a nontricial matrix $U=\eta\sigma_2$ to flip the spin. Note that $T^2=\eta\sigma_2 K\eta \sigma_2 K = \eta\sigma_2\eta^*\sigma_2^*KK=-1$. This is the origin of Kramer's degeneracy

My question is that, it seems we can choose a different choice of the time reversal to be $U=\eta\sigma_1$ equally well. This also clearly flips the spin-up state to the spin-down state and vice versa. With this choice, we have
$$T^2=\eta\sigma_1 K\eta \sigma_1 K = \eta\sigma_1\eta^*\sigma_1^*KK=1$$
because $\sigma_1$ is real. We won't see the Kramer degeneracy anymore. But physics should not depend on the choice. Is there any other constrains we have to impose in order to get $U=\eta\sigma_2$?
 A: We know that the action of time-reversal on $\sigma_a$ is
\begin{equation}
T\sigma_aT^{-1}=-\sigma_a.
\end{equation}
Now, we have that
\begin{equation}
T\sigma_aT^{-1}=(UK)\sigma_a (K U^{-1})=U \sigma_a^{*}U^{-1}=(-1)^{a+1}U\sigma_aU^{-1}, \quad (a=1,2,3).
\end{equation}
which thus means that $U$ must satisfy
\begin{equation}
U\sigma_a U^{-1}=(-1)^a\sigma_a.
\end{equation}
If we write $U=e^{i d \sum_a n_a\sigma_a},$ where $d>0$ and $\sum_a n^2_a=1,$ then we have that
\begin{equation}
    U\sigma_a U^{-1}=\sigma_a\cos 2d+\sum_{bc}\epsilon_{abc}n_b \sigma_c \sin 2d+n_a\sum_{b}n_b\sigma_b(1-\cos 2d).
\end{equation}
This form of the transformation allows us to obtain a set of equations for $d,n_a$ by evaluating the trace
\begin{equation}
\text{Tr}\left[\sigma_b U\sigma_a U^{-1}\right]=2(-1)^{a}\delta_{ab}.
\end{equation}
Inspection of these equations lead to the only solution $n_a=\delta_{a2}$ and $d=\pi/2.$ We thus conclude that
\begin{equation}
    T=e^{i\frac{\pi}{2}\sigma_2}K=i\sigma_2 K.
\end{equation}
