Visualizing the wave operator in two dimensions

Question

For $n\geq 1$, let $D_n$ be the Dirac operator on the spinor bundle on the $n$-dimensional sphere $S^n$. For example, $D_1$ acts on the trivial bundle $S^1\times\mathbb{C}\to S^1$, and can be explicitly written as $-i\frac{d}{d\theta}$.

Let $e^{itD_n}$ be the wave operator associated to $D_n$; that is, the operator that solves the wave equation $$\frac{du}{dt}=iD_nu,$$ where $u$ is an $L^2$-spinor.

In the case of $n=1$, $e^{itD_n}$ is just the operator that "shifts" everything on the circle by $t$ in the $\theta$-direction.

Question 1: Let $D_{\mathbb{R}^2}$ be the Dirac operator on $\mathbb{R}^2$. How might one visualize $e^{itD_{\mathbb{R}^2}}$?

Question 2: How might one visualize $e^{itD_2}$ on $S^2$?

Answer 1

The Dirac operator on $\mathbb{R}^2$, $$ D=\begin{pmatrix}0&k_x-ik_y\\ k_x+ik_y&0\end{pmatrix},\;\;\mathbf{k}=-i\frac{\partial}{\partial \mathbf{r}},$$ acting as $e^{-iDt}$ on a spinor plane wave $e^{i\mathbf{q}\cdot\mathbf{r}}\psi$, rotates the spinor $\psi$ on the Bloch sphere, by an angle $\theta=\tfrac{1}{2}t|\mathbf{q}|$ around the axis parallel to the vector $\mathbf{q}$.