We now consider the improper Lorentz transformation of reflection in space or the parity transformation:

(5.149) |

We need to solve (5.122) for

(5.150) |

We denote the Lorentz operator by for parity. Consider the following ansatz

(5.151) |

where is an arbitrary phase. Using equation 5.122, we have

(5.152) |

as required.

In analogy to the proper Lorentz transformation for which a rotation of reproduces the original spinors, we postulate that four space-inversions will reproduce the original spinors.

(5.153) |

Therefore

(5.154) |

We see that

(5.155) | |||

(5.156) |

and is unitary.

The wave function thus transforms as

(5.157) |

In the nonrelativisitc limit approaches an eigenstate of . The positive- and negative-energy states at rest have opposite eigenvalues, or intrinsic parities. The intrinsic parity of a Dirac particle and antiparticle are opposite. This is to be contrasted to the Klein-Gordon case wherein one finds identical parities for the particle and antiparticle solutions.

2004-03-18