Besides plane-wave solutions of definite momentum, we can have solutions which are the Fourier transform of which depend only on momentum:

The relativistic invariance of solutions of this form is not immediately obvious.

The general solution to the Klein-Gordon equation has either positive or negative energy. They can be written (using ) as a Fourier transform of ,

where the index is used to label different solutions of identical mass. This form is manifestly Lorentz invariant. The Lorentz invariant step function is

(4.25) |

Lorentz invariance of restricts to be a time-like vector and thus distinguishes between past and future. Thus the expression ensures the condition .

We can show that equation 4.24 gives rise to the usual form of the solutions by rewriting the general solution, using

(4.26) |

and then applying the identity equation 2.28.

For positive energy

(4.27) |

For negative energy

(4.29) |

Equations 4.28 and 4.30 are identical to the previously solutions (equation 4.23).

2004-03-18