Just a brief reminder that what we do is not as difficult as the math for particle physics

Quarks and Leptons:

An Indrodoctory Course In Modern Particle Physics

 Francis Halzen and Alan D. Martin

 

 Green's  Functions

 

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

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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

 

                                               (6.120)

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).