We have just seen how one can create a state of arbitrary momentum by Lorentz transforming the solution for a particle at rest. In a similar manner, we can create a state of arbitrary spin polarized along the direction by applying the rotation operator,

(5.198) |

to the solution for a particle at rest and polarized in the direction. The defining relationship for such a state is

(5.199) |

where the spinor corresponds to a particle polarized along the direction of the unit vector . Remember that the spin operator is and thus the eigenvalue of the spin operator in this case is really , as we might expect.

Let denote a spinor of positive energy, momentum , and spin . The spin vector is defined as

(5.200) |

where is the polarization unit vector in the rest frame and is a Lorentz transformation from the rest frame. Thus we also have , where . This tells us that

(5.201) |

and

(5.202) |

which is true in any frame since they are Lorentz scalars. Thus in the rest frame

(5.203) |

Let denote a spinor of negative energy, with polarization in the reset frame. In this case,

(5.204) |

Therefore in an arbitrary Lorentz frame

where is a 4-vector, which in the rest frame is . An arbitrary spinor is thus specified by the sign of the energy, its momentum , and polarization in the rest frame .

2004-03-18