Pauli Matrices & 2D Rotation Operators?

Question

I was doing a strange calculation with my teacher the other day: find the eigenvalues and eigenvectors of the 2D rotation operator. Intuitively, there should be no solution to this problem in $\mathbb{R}^2$. However, when we extend to $\mathbb{C}^2$, we find that the unnormalized eigenvectors are:

$$ \vec{v}_1 = (1,i)$$ $$\vec{v}_2 = (1,-i) $$

Then we realized that these vectors are exactly the eigenvectors of one of the Pauli Matrices $\sigma_y$:

$$\left( \begin{array}{cc} 0 & -i \\ i & 0 \\ \end{array} \right)$$

What do we make of that? What happens in other dimensions? Is 2D very special?

Answer 1

Well, it is trivial. Real $2D$ rotations, viewed both as matrices in $\mathbb R^2$ or $\mathbb C^2$ are all of the form $$R(\theta)= e^{\theta i\sigma_2}\:.$$ (Notice that $i\sigma_2$ is real antisymmetric as it must be it being a generator of $so(2)$.) Moreover from the spectral decomposition in $\mathbb C^2$, $e^{\theta i\sigma_2}$ has the same eigenvectors as $\sigma_2$.

Answer 2

To answer the generic dimension part, in case it is not self-evident from Valter's answer:

In D dimensions, the rotation matrix is the exponential of an angle θ times a matrix K, a normalized generator of the corresponding rotation group SO(D) around some unit axis D-vector k, in the vector representation, so the matrix is D×D. The eigenvectors of these matrices K will likewise be the eigenvectors of the rotation operator.

However, since there are several (an infinity) axes of rotation for D>2, there will be an infinity of eigenvector sets, each characterized by the specific spin matrix K.

For instance, in D=3, the vector representation, the rotation matrix is given as this exponential of the cross-product matrix $$ \mathbf{K}= \left[\begin{array}{ccc} 0 & -k_3 & k_2 \\ k_3 & 0 & -k_1 \\ -k_2 & k_1 & 0 \end{array}\right],$$ expanded trivially by the Rodrigues formula to just $\mathbf{R} = \mathbf{I} + (\sin\theta) \mathbf{K} + (1-\cos\theta)\mathbf{K}^2 ~$. Thus the three eigenvectors of K (one of which is the null vector k) are also the eigenvectors of R.