Consider time-reversal invariance, . This time we start with the Dirac equation in Hamiltonian form

(5.253) |

Define the transformation such that and , we have

(5.254) |

Since is generated by currents which reverse sign when the sense of time is reversed,

(5.255) | |||

(5.256) |

Also , since . The transformation must cause to get the correct form, therefore can be defined as:

- take complex conjugate,
- multiply by constant matrix .

(5.257) |

Therefore

(5.258) |

This implies must commute with and and anticommute with and . Therefore we can try

(5.259) |

The phase factor is arbitrary.

We apply to a plane-wave solution for a free particle of positive energy. Since

(5.260) |

and

(5.261) |

where ; Therefore projects out a free-particle solution with reversed direction of momentum and spin . This is known as ``Wigner time reversal''.

2004-03-18