Maxwell's equations of classical electrodynamics in vacuum are

(7.88) | |||

where we are using Heaviside-Lorentz (rationalized Gaussian) units.

Defining and , the four-vector potential is related to the electric and magnetic fields by

(7.89) |

Further, we can show that in terms of the antisymmetric field strength tensor

(7.90) |

that Maxwell's equations now take the compact form

(7.91) |

and current conservation, , follows as a natural compatibility condition.

These equations are equivalent to the following covariant equation for .

(7.92) |

which is general and is in the presence of charge and currents.

and are unchanged by the gauge transformation

(7.93) |

where is any scaler function of . We use this freedom to pick so that we can write Maxwell's equations in the form

(7.94) |

The requirement is known as the Lorentz condition. The process of choosing a particular condition on so as to define it uniquely is called ``choosing a gauge''. Potentials satisfying are said to be ``in the Lorentz gauge''. However, even after choosing a gauge, there is still some residual freedom in the choice of the potential . We can still make another gauge transformation

(7.95) |

where is any function that satisfies

(7.96) |

This last equation ensures that the Lorentz condition is still satisfied.

2004-03-18