Propagator theory is based on the Green's function method of solving inhomogeneous differential equations. We explain the method in terms of a single example.

Suppose we wish to solve Poisson's equation

(6.1) |

for a known charge distribution , subject to some boundary conditions. It is easier to first solve the ``unit source'' problem

(6.2) |

where is the potential at due to a unit source at . We then move this source over the charge distribution and accumulate the total potential at from all possible volume elements :

(6.3) |

We can check directly that is the desired solution.

(6.4) | |||

In the case of the electron propagator, appears on both sides of the solution, and so an iterative perturbation series-solution in powers of is required.

2004-03-18