The Dirac equation for an electron on a curved space-time may be viewed as an eigenvalue problem for the Dirac operator on the spinor fields of the space-time. A general eigenvalue problem for the Dirac operator on a metric manifold M in terms of spinor and tangent fields defined via the Clifford algebra is derived herein. Then it is shown how to solve the Dirac eigenvalue problem on an imbedded submanifold N, of codimension one in M, by solving an eigenvalue equation on M. This is applied to the case of a sphere or pseudosphere imbedded in flat space. Eigenvalues and eigenfunctions for the Dirac operator on any sphere or pseudosphere are determined. In particular, when the pseudosphere is a space-time, the Dirac equation for a free lepton in this space-time can be solved. The resulting mass spectrum is discrete and depends on the curvature of the space-time.
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Mathematical Physics