EIGENVALUES AND EIGENFUNCTIONS OF THE DIRAC OPERATOR ON SPHERES AND PSEUDOSPHERES

The Dirac equation for an electron on a curved space-time may be viewe d as an eigenvalue problem for the Dirac operator on the spinor fields of the space-time. A general eigenvalue problem for the Dirac operato r 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 s olve the Dirac eigenvalue problem on an imbedded submanifold N, of cod imension one in M, by solving an eigenvalue equation on M. This is app lied to the case of a sphere or pseudosphere imbedded in flat space. E igenvalues 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 c an be solved. The resulting mass spectrum is discrete and depends on t he curvature of the space-time.