Clifford algebra

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

J. P. Crawford

Research output: Contribution to journalArticle

22 Citations (Scopus)

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 languageEnglish (US)
Pages (from-to)576-583
Number of pages8
JournalJournal of Mathematical Physics
Volume32
Issue number3
DOIs
StatePublished - Jan 1 1991

Fingerprint

Clifford Algebra
Spinor
Poincaré
algebra
Metric
Automorphism Group
Normalization
Odd
Paul Adrien Maurice Dirac
Lorentz Group
Invariant Metric
Isomorphism
Subgroup
isomorphism
Distinct
subgroups
Arbitrary
preserving

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

@article{b1343eb062ad4415983e5ec42040e0dc,
title = "Clifford algebra: Notes on the spinor metric and Lorentz, Poincare, and conformal groups",
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.",
author = "Crawford, {J. P.}",
year = "1991",
month = "1",
day = "1",
doi = "10.1063/1.529397",
language = "English (US)",
volume = "32",
pages = "576--583",
journal = "Journal of Mathematical Physics",
issn = "0022-2488",
publisher = "American Institute of Physics Publising LLC",
number = "3",

}

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

In: Journal of Mathematical Physics, Vol. 32, No. 3, 01.01.1991, p. 576-583.

Research output: Contribution to journalArticle

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.

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

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 -