Disclaimer: I am not a physicist; I am a geometer (and a student!) trying to learn some physics. Please be gentle. Thanks!
When solving the Schrödinger equation for a particle in a spherical potential, it seems common to separate variables into angular and radial components. The angular evolution can then be expressed in terms of eigenfunctions of the Laplace-Beltrami operator $\Delta$ on the sphere, i.e., the spherical harmonics. (It is my understanding that these eigenfunctions or eigenstates also have some physical significance, namely that eigenfunctions with the same eigenvalue correspond to states of equal energy.)
When solving the Dirac equation (again with a spherical potential) you'd expect a similar story: separate into angular and radial components and write the angular evolution in terms of the eigenfunctions of the Riemannian Dirac operator $D$ on the sphere. And, you'd expect these eigenfunctions would have a similar physical interpretation to the non-relativistic case (after all, the only thing we changed was the energy-momentum relationship).
However, the references I'm finding on the Dirac equation with central potential write solutions in terms of the spherical spinors $\Omega$, which are themselves simple functions of the spherical harmonics $Y_l^m$. This situtation seems odd to me because, although eigenfunctions of $D$ are also eigenfunctions of $\Delta$, the opposite is not true. In particular, $D$ will have both positive and negative eigenvalues, and so eigenspaces with equal value but opposite sign get "mixed" when we square $D$ (recall that on, say, Euclidean $R^3$, $D^2=\Delta$). I'm not sure about the physical interpretation, though, because I don't understand the physical meaning of eigenfunctions of the Dirac operator.
Here are some more concrete questions:
- what do eigenfunctions of the $D$ represent physically?
- why are the spherical harmonics used for separation of variables rather than eigenfunctions of $D$?
- alternatively, are there cases where eigenfunctions of $D$ are used to solve Dirac's equation?
Pedagogical references are appreciated. Thanks!