We go to the rest frame and try to find a projection operator in a covariant form. A candidate for a spin-up particle is , where is the third Pauli-spin matrix. Removing the explicit dependence we can write

(5.212) |

where is a unit 3-vector. Extending the operator to 4-dimensions in the rest frame we have

(5.213) |

Because we are in the rest frame acting upon the Dirac spinor becomes and we can deal with this overall sign later. The covariant Dirac spin projection operator has the form

(5.214) |

Notice that and that they are different operators. For a general spin vector we have

(5.215) |

In the reset frame

(5.216) | |||

(5.217) | |||

(5.218) | |||

(5.219) |

The physical motivation for this apparent backward association of the eigenvalues for the negative-energy solution will appear when we come to the hole theory.

Using the definitions (5.206) we can write

(5.220) |

Because of the covariant form of the projection operator, we may write for any polarization vector

(5.221) |

We now show that the spin projection operators have the projection operator properties:

(5.222) |

and

(5.223) |

also

(5.224) |

The energy and spin projection operators commute:

(5.225) |

2004-03-18