We now cast the Dirac equation into a more apparent covariant form. Multiplying (5.3) by and defining

(5.96) |

where , we have

(5.97) |

where

(5.98) |

In terms of the momentum operator we write

(5.99) |

Introducing the Feynman dagger, or slash notation, for 4-vector , we have

(5.100) |

Also notice that

(5.101) |

We write

(5.102) |

We introduce the electromagnetic interaction by the usual minimal substitution

(5.103) |

Let us study the properties of the matrices.

(5.104) |

And

(5.105) |

Since

(5.106) |

Therefore

where the matrices are and . Although the Dirac matrices are written with Greek indices, they are not four vectors. Rather, they have the same value in every frame.

The matrices are anti-hermitian and is hermitian:

(5.108) | |||

(5.109) |

This can be summarized by writing

(5.110) |

Using our previous representation (equation 5.10),

(5.111) | |||

(5.112) |

2004-03-18