### Abstract

The Dirac equation is most easily formulated in terms of functions from space-time to the Clifford algebra R_{3,1} and the gradient operator. In this setting we may construct wave packet solutions for all leptons by introducing a single operator which transforms a real valued function on the group Spin^{+}(3,1) to a wave function by using integration on the group itself. Applying the operator to a certain space of real valued functions on the group, we produce Spin^{+}(3,1)-invariant solution spaces which may then be classified. The results are one space each for electrons and positrons and two parameter families of spaces for neutrinos and antineutrinos. The electron and neutrino spaces display an asymptotic symmetry at high energies, as do the positron and antineutrino spaces. If we also insist that the solution spaces be translation invariant, then we get the familiar two-component neutrino theory and the asymptotic symmetry between leptons of the same handedness used in the theory of weak interactions. This symmetry is purely a result of the Dirac theory in the Clifford algebra setting.

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

Pages (from-to) | 153-165 |

Number of pages | 13 |

Journal | Reports on Mathematical Physics |

Volume | 32 |

Issue number | 2 |

DOIs | |

State | Published - 1993 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mathematical Physics
- Statistical and Nonlinear Physics

### Cite this

**Orbits of Spin ^{+}(3,1), wave packets and asymptotic lepton symmetries.** / Clarkson, Eric W.

Research output: Contribution to journal › Article

^{+}(3,1), wave packets and asymptotic lepton symmetries',

*Reports on Mathematical Physics*, vol. 32, no. 2, pp. 153-165. https://doi.org/10.1016/0034-4877(93)90011-3

}

TY - JOUR

T1 - Orbits of Spin+(3,1), wave packets and asymptotic lepton symmetries

AU - Clarkson, Eric W

PY - 1993

Y1 - 1993

N2 - The Dirac equation is most easily formulated in terms of functions from space-time to the Clifford algebra R3,1 and the gradient operator. In this setting we may construct wave packet solutions for all leptons by introducing a single operator which transforms a real valued function on the group Spin+(3,1) to a wave function by using integration on the group itself. Applying the operator to a certain space of real valued functions on the group, we produce Spin+(3,1)-invariant solution spaces which may then be classified. The results are one space each for electrons and positrons and two parameter families of spaces for neutrinos and antineutrinos. The electron and neutrino spaces display an asymptotic symmetry at high energies, as do the positron and antineutrino spaces. If we also insist that the solution spaces be translation invariant, then we get the familiar two-component neutrino theory and the asymptotic symmetry between leptons of the same handedness used in the theory of weak interactions. This symmetry is purely a result of the Dirac theory in the Clifford algebra setting.

AB - The Dirac equation is most easily formulated in terms of functions from space-time to the Clifford algebra R3,1 and the gradient operator. In this setting we may construct wave packet solutions for all leptons by introducing a single operator which transforms a real valued function on the group Spin+(3,1) to a wave function by using integration on the group itself. Applying the operator to a certain space of real valued functions on the group, we produce Spin+(3,1)-invariant solution spaces which may then be classified. The results are one space each for electrons and positrons and two parameter families of spaces for neutrinos and antineutrinos. The electron and neutrino spaces display an asymptotic symmetry at high energies, as do the positron and antineutrino spaces. If we also insist that the solution spaces be translation invariant, then we get the familiar two-component neutrino theory and the asymptotic symmetry between leptons of the same handedness used in the theory of weak interactions. This symmetry is purely a result of the Dirac theory in the Clifford algebra setting.

UR - http://www.scopus.com/inward/record.url?scp=43949168212&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=43949168212&partnerID=8YFLogxK

U2 - 10.1016/0034-4877(93)90011-3

DO - 10.1016/0034-4877(93)90011-3

M3 - Article

AN - SCOPUS:43949168212

VL - 32

SP - 153

EP - 165

JO - Reports on Mathematical Physics

JF - Reports on Mathematical Physics

SN - 0034-4877

IS - 2

ER -