# Time-reversal symmetry and the generalized special axes (eg: $y$) in any $D$ space dimension

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?

2.

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?

• 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 $\sigma_{2}$.
– Buzz
Oct 4 '20 at 23:15