The Dirac oscillator and local automorphism invariance

James P. Crawford

Research output: Contribution to journalArticlepeer-review

8 Scopus citations


The Dirac oscillator is a relativistic generalization of the quantum harmonic oscillator. In particular, the square of the Hamiltonian for the Dirac oscillator yields the Klein-Gordon equation with a potential of the form (ar2+bL·S), where a and b are constants. To obtain the Dirac oscillator, a "minimal substitution" is made in the Dirac equation, where the ordinary derivative is replaced with a covariant derivative. However, a very unusual feature of the covariant derivative in this case is that the potential is a nontrivial element of the Clifford algebra. A theory which naturally gives rise to gauge potentials which are nontrivial elements of the Clifford algebra is that based on local automorphism invariance. An exact solution of the pure automorphism gauge field equations which reproduces both the potential term and the mass term of the Dirac oscillator is presented herein.

Original languageEnglish (US)
Pages (from-to)4428-4435
Number of pages8
JournalJournal of Mathematical Physics
Issue number10
StatePublished - 1993

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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