1. In 3D space, it is common to choose the time-reversal symmetry acting on spin-1/2 doublet fermions as $$ T = i \sigma_y K = \begin{pmatrix} 0 & 1\\ -1& 0\end{pmatrix} $$ where $K$ is complex conjugation and $\sigma_j$ is a rank-2 Pauli matrix. Alternatively, we can write it as a $\theta=\pi$ rotation along the $y$ axis $$ T = \exp(i \theta S_y ) =\exp\left ( i \frac{\pi}{2}\sigma_y\right ). $$ See Wikipedia notation as $T = e^{-i\pi J_y/\hbar} K$ https://en.wikipedia.org/wiki/T-symmetry#Anti-unitary_representation_of_time_reversal.

question 1: why do we have a special $y$ axis picked up among the three $x,y,z$? I suppose it has something to do with the fact that

  • the complex conjugation $\sigma_x^*=\sigma_x$ and $\sigma_z^*=\sigma_z$ but $\sigma_y^*=-\sigma_y$

  • the symmetric matrix $\sigma_x^T=\sigma_x$ and $\sigma_z^T=\sigma_z$ but anti-symmetric matrix $\sigma_y^T=-\sigma_y$

  • also, $\sigma_y \sigma_a \sigma_y = -\sigma_a^*$ for any $a$ axis.

But can we precisely spell out the fact why $y$ axis is special?


question 2: If we generalize the time-reversal $T$ formula to other $D$ space dimensions, what the $T=?$ Do we still pick up certain special directions (those have Lie algebra generator matrix representations to be complex conjugation to its minus sign and anti-symmetric matrix)?

can we write down this generalized time-reversal $T$ formula in any $D$ space dimensions? by what principles?

  • 3
    $\begingroup$ There is nothing special about $y$, except that we have chosen, by convention, to make the $y$-axis the one that corresponds to the imaginary Pauli matrix $\sigma_{2}$. $\endgroup$
    – Buzz
    Oct 4 '20 at 23:15

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