Quarks and Leptons:
An Indrodoctory Course In Modern Particle Physics
Francis Halzen and Alan D. Martin
Propagator theory is based on the Green's function method of solving inhomoge neous differential equations. We explain the method in terms of a simple example. Suppose we wish to solve Poisson's equation
146 Electrodynamics of Spin-} Particles
Fig. 6.14 G is the potential at x due to a unit
superposition to obtain the cumulative potential at x, (6.121), arising from all possible elemental charges pd 3x'.
for a known charge distribution p(x), subject to some boundary condition. It is easier to first solve the "unit source" problem
where G(x, x') is the potential at x due to a unit source at x'. [For the boundary condition that G --. 0 at large distances, it is easy to show that G = 1/ (4'1TIX- x'l)]. We then move this source over the charge distribution and accu mulate the total potential at x from all possible volume elements d 3x':
q,(x) = j G(x,x') p(x') d 3x', (6.121)
see Fig. 6.14. We can check directly that q, is the desired solution of (6.119) by operating with V 2 on (6.121).