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!
Best Answer
There might be a small bit of confusion in the question here.
[Skip all this stuff below until the next bracketed comment.]
First recall that $\mathcal{so}(3)$ has representations that integrate to representations of $SU(2)$ rather than $SO(3)$, and these are the "half-integral" spin. Likewise for $\mathcal{so}(3,1)$ and $SU(2)\times SU(2)$ and $SO(3,1)$, respectively.
Now for a manifold with Lorentzian metric, a representation of $\mathcal{so}(3,1)$ indicates what kind of particle/object/tensor/spinor you're talking about. The representation tells you how to transform the object/section when you change coordinates. (N.B.: since we're talking about bundles associated to the frame bundle, changing coordinates induces a change of trivialization of the bundle. Ordinarily, the two concepts are indpendent.) More specifically, tensors correspond to "integral spin" representations, e.g. the 4-dimensional representation is vectors, while the 6-dimensional anti-symmetric representation describes two-forms (such as field strengths).
So spinors are sections of bundles which correspond to representations of $\mathcal{so}(3,1)$ which integrate to a representation of $SU(2)\times SU(2)$. Practically speaking, in order to covariantly differentiate such an object, you can do the following. The Levi-Civita connection already gives you an element of $\mathcal{so}(3,1)$ (a matrix) for each tangent index. Plus you have a representation of $\mathcal{so}(3,1)$, so you act with this element of $\mathcal{so}(3,1)$ by your representation. This is what the gamma matrices are about.
[Oy, this is getting too long.]
Now here's the thing: the spinor representations decompose into different irreducible components, and the Dirac operator maps one representation (positive chirality) into another (negative chirality)! That is, it maps sections of one spinor bundle into sections of another. You can of course look at the Dirac operator on the sum of these bundles, but eigenvectors do not have an evident physical interpretation (they are of mixed chirality). In flat space, the square of the Dirac operator is a multiple of the identity endomorphism on the bundle, so eigenvalues make perfect sense and can be written in terms of functions times the (global) basis elements for spinors.