### Abstract

A particular normalization for the set of basis elements {i} of the complex Clifford algebras C(p,q) is motivated and defined by demanding that the physical bispinor densities i=i be real. This condition, referred to here as Dirac normalization, also necessitates the introduction of the spinor metric, and the solution of the metric conditions is given for arbitrary (p,q); when N=p+q is even the metric is unique, and when N is odd there are two distinct metrics. Then the Dirac normalization preserving automorphism group of the basis is explored. This is also the group of transformations leaving the spinor metric invariant. In particular, the physically important cases of the Lorentz, Poincare, and conformal groups are sought as subgroups of the automorphism group. As expected, it is found that the Lorentz group is always contained in the automorphism group. However, it is found that the Poincare and conformal groups are contained only in the cases where N is even and q is odd. Furthermore, when N is odd these groups may be found in the full isomorphism group, but only for one of the two possible spinor metrics. Possible physical implications of these results are discussed.

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

Pages (from-to) | 576-583 |

Number of pages | 8 |

Journal | Journal of Mathematical Physics |

Volume | 32 |

Issue number | 3 |

DOIs | |

State | Published - Jan 1 1991 |

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### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Journal of Mathematical Physics*,

*32*(3), 576-583. https://doi.org/10.1063/1.529397

}

*Journal of Mathematical Physics*, vol. 32, no. 3, pp. 576-583. https://doi.org/10.1063/1.529397

**Clifford algebra : Notes on the spinor metric and Lorentz, Poincare, and conformal groups.** / Crawford, J. P.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Clifford algebra

T2 - Notes on the spinor metric and Lorentz, Poincare, and conformal groups

AU - Crawford, J. P.

PY - 1991/1/1

Y1 - 1991/1/1

N2 - A particular normalization for the set of basis elements {i} of the complex Clifford algebras C(p,q) is motivated and defined by demanding that the physical bispinor densities i=i be real. This condition, referred to here as Dirac normalization, also necessitates the introduction of the spinor metric, and the solution of the metric conditions is given for arbitrary (p,q); when N=p+q is even the metric is unique, and when N is odd there are two distinct metrics. Then the Dirac normalization preserving automorphism group of the basis is explored. This is also the group of transformations leaving the spinor metric invariant. In particular, the physically important cases of the Lorentz, Poincare, and conformal groups are sought as subgroups of the automorphism group. As expected, it is found that the Lorentz group is always contained in the automorphism group. However, it is found that the Poincare and conformal groups are contained only in the cases where N is even and q is odd. Furthermore, when N is odd these groups may be found in the full isomorphism group, but only for one of the two possible spinor metrics. Possible physical implications of these results are discussed.

AB - A particular normalization for the set of basis elements {i} of the complex Clifford algebras C(p,q) is motivated and defined by demanding that the physical bispinor densities i=i be real. This condition, referred to here as Dirac normalization, also necessitates the introduction of the spinor metric, and the solution of the metric conditions is given for arbitrary (p,q); when N=p+q is even the metric is unique, and when N is odd there are two distinct metrics. Then the Dirac normalization preserving automorphism group of the basis is explored. This is also the group of transformations leaving the spinor metric invariant. In particular, the physically important cases of the Lorentz, Poincare, and conformal groups are sought as subgroups of the automorphism group. As expected, it is found that the Lorentz group is always contained in the automorphism group. However, it is found that the Poincare and conformal groups are contained only in the cases where N is even and q is odd. Furthermore, when N is odd these groups may be found in the full isomorphism group, but only for one of the two possible spinor metrics. Possible physical implications of these results are discussed.

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UR - http://www.scopus.com/inward/citedby.url?scp=0040358658&partnerID=8YFLogxK

U2 - 10.1063/1.529397

DO - 10.1063/1.529397

M3 - Article

VL - 32

SP - 576

EP - 583

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 3

ER -