Eigenvalues and eigenfunctions of the Dirac operator on spheres and pseudospheres

Research output: Contribution to journalArticle

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 languageEnglish (US)
Pages (from-to)2064-2073
Number of pages10
JournalJournal of Mathematical Physics
Volume35
Issue number5
StatePublished - 1994
Externally publishedYes

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Pseudosphere
Eigenvalues and Eigenfunctions
Dirac Operator
Eigenvalues and eigenfunctions
eigenvectors
eigenvalues
Space-time
operators
Eigenvalue Problem
Algebra
Spinor
Dirac Equation
Dirac equation
Electrons
Clifford Algebra
Tangent line
Submanifolds
Codimension
Paul Adrien Maurice Dirac
tangents

ASJC Scopus subject areas

  • Organic Chemistry

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

In: Journal of Mathematical Physics, Vol. 35, No. 5, 1994, p. 2064-2073.

Research output: Contribution to journalArticle

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