### Abstract

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.

Original language | English (US) |
---|---|

Pages (from-to) | 2064-2073 |

Number of pages | 10 |

Journal | Journal of Mathematical Physics |

Volume | 35 |

Issue number | 5 |

State | Published - 1994 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Organic Chemistry

### Cite this

**Eigenvalues and eigenfunctions of the Dirac operator on spheres and pseudospheres.** / Clarkson, Eric W.

Research output: Contribution to journal › Article

*Journal of Mathematical Physics*, vol. 35, no. 5, pp. 2064-2073.

}

TY - JOUR

T1 - Eigenvalues and eigenfunctions of the Dirac operator on spheres and pseudospheres

AU - Clarkson, Eric W

PY - 1994

Y1 - 1994

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=36449002481&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=36449002481&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:36449002481

VL - 35

SP - 2064

EP - 2073

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 5

ER -