### Abstract

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 (ar^{2}+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 language | English (US) |
---|---|

Pages (from-to) | 4428-4435 |

Number of pages | 8 |

Journal | Journal of Mathematical Physics |

Volume | 34 |

Issue number | 10 |

DOIs | |

State | Published - Jan 1 1993 |

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### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Journal of Mathematical Physics*,

*34*(10), 4428-4435. https://doi.org/10.1063/1.530348

}

*Journal of Mathematical Physics*, vol. 34, no. 10, pp. 4428-4435. https://doi.org/10.1063/1.530348

**The Dirac oscillator and local automorphism invariance.** / Crawford, James P.

Research output: Contribution to journal › Article

TY - JOUR

T1 - The Dirac oscillator and local automorphism invariance

AU - Crawford, James P.

PY - 1993/1/1

Y1 - 1993/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=21344483231&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=21344483231&partnerID=8YFLogxK

U2 - 10.1063/1.530348

DO - 10.1063/1.530348

M3 - Article

AN - SCOPUS:21344483231

VL - 34

SP - 4428

EP - 4435

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 10

ER -