Visualizing the wave operator in two dimensions

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 $$S^2$$?

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 $$\mathbf{q}$$.
• Hmm I'm not so familiar with the language in your answer (eg. the Bloch sphere), so I have some naive questions. In particular, can I think of $\psi$ as, say, an $L^2$-spinor? And why consider $e^{i\textbf{q}\cdot\textbf{r}}\psi$ instead of just $\psi$? Thanks! Apr 6 '21 at 22:26