The relativistic propagator, is defined to satisfy a Green's function equation

(6.64) |

The propagator is a matrix corresponding to the dimensionality of matrices. Note that the definition of the relativistic propagator differs from the nonrelativistic counterpart: the differential operator occurring in the nonrelativistic equation has been multiplied by in the relativistic equation in order to form the covariant operator .

Suppressing the indices we have

We can compute the free-particle propagator, , using

(6.66) |

by Fourier transforming to momentum space. depends only on the interval . This property is a manifestation of the homogeneity of space and time, and is general would not be valid for the interacting propagator . We have

(6.67) |

This gives

(6.68) |

Solving for the Fourier amplitude and reverting to the matrix shorthand, we find

(6.69) |

A prescription for how to handle the singularity at or ( ) is needed. This comes from the boundary conditions put on in the integration.

The interpretation given to the Green's function is that it represents the wave produced at the point by a unit source located at the point . The Fourier components of such a localized point source contain many momenta larger than , the reciprocal of the electron Compton wavelength, and we expect that positrons as well as electrons may be created at by the source. However, a necessary physical requirement of the hole theory is that the wave propagating from into the future consist only of positive-energy electron and positron components. Since positive-energy positrons and electrons are represented by wave functions with positive frequency time behaviour can contain in the future , only positive-frequency components.

We perform the integration along the contour in the complex -plane. For , the contour is closed in the lower half-plane and includes the positive-frequency pole at only. This gives

(6.70) |

so that the wave at contains positive-frequency components only. For , the contour can be closed above, including the pole at . This gives

(6.71) |

showing the propagator to consist of negative-frequency waves for . Any other choice of contour leads to negative-energy waves propagating into the future or positive-energy waves propagating into the past. Moreover, the negative-energy waves propagating into the past that we have just found are welcome; they are the positive-energy positrons. The origin of the negative-energy waves is the pole at , which was not present in the nonrelativistic theory.

The choice of the contour is summarised by adding a small positive imaginary part to the denominator, or simply taking , where the limit is understood:

(6.72) |

The two integrations can be combined by introducing projection operators and changing to in the negative-frequency part:

(6.73) |

with .

Equivalently, writing for normalized plane-wave solutions, we find

(6.74) |

We see that carries the positive-energy solutions forward in time and the negative-energy ones backward in time

The minus sign in the second equation results from the difference of the direction of propagation in time between (6.75) and (6.76).

is known as the Feynman propagator. This spin-1/2 propagator is related to the Klein-Gordon propogator by

(6.77) |

From the free propagator we may formally construct the complete Green's function and the -matrix elements, that is, the amplitudes for various scattering processes of electrons and positrons in the presence of force fields. The exact Feynman propagator satisfies (6.65) and can be expressed in terms of a superposition of free Feynman propagators

This is the relativistic counterpart of the Lippmann-Schwinger equation. Another notation for is . This integral equation determines the complete propagator in terms of the free-particle propagator .

Proceeding in analogy to the nonrelativistic treatment, the iteration of the integral equation yields the following multiple scattering expansion

(6.80) | |||

Equation 6.78 can be viewed as an inhomogeneous Dirac equation of the form which is solved by the Green's function techniques as follows

(6.81) |

The exact solution of the Dirac equation with the Feynman boundary conditions, is

where is the free-particle solution. The scattering wave contains only positive frequencies in the future and negative frequencies in the past

(6.83) | |||

(6.84) |

2004-03-18