Single-Particle Electrodynamics - Assassination Science

as a mathematical aid—it not being, in itself, a physically observable quantity
in classical physics. In terms of the A µ , the Maxwell equation (5.7) is
∂ 2 A µ − ∂ µ (∂ ν A ν ) = J µ . (5.8)
It now simplifies the computations to choose a Lorentz gauge, in which
∂ ν A ν LG = 0. (5.9)
This choice, of course, destroys gauge invariance, so we should aim to return
to the gauge-invariant (and physically meaningful) field strength tensor
F (x) as soon as the possibility arises. With the choice (5.9), equation (5.8)
becomes
∂ 2 A µ LG = J µ . (5.10)
The utility of the choice (5.9) lies with the fact that (5.10) now represents
four uncoupled differential equations in Minkowski space, one for each µ =
0, 1, 2, 3. It therefore suffices to find a Green function D(x, x ′ ) satisfying the
equation
∂ 2 xD(x, x ′ ) = δ (4) (x − x ′ ). (5.11)
Now, this is a merely mathematical problem; the standard textbook solution
(see, e.g., [113, pp. 609–11] for a derivation) is
D r (x − x ′ ) = 1
2π ϑ(x0 − x ′ 0 ) δ [ (x − x ′ ) 2] , (5.12)
where we have selected the retarded solution, and ϑ(t) is the Heaviside step
function of Section A.5.3. Using this Green function, the solution to (5.10)
can be written down directly:
∫
A µ LG(x) = A µ in(x) + d 4 x ′ D r (x − x ′ )J µ (x ′ ), (5.13)
where A µ in(x) represents the Lorentz-gauge four-potential of any radiation
“incoming”, that is not due to the particle in question. Since the equations
are linear, we can omit the term A µ in(x) from our explicit considerations, it
being understood that the incoming fields may simply be added to the fields
generated by the particle in question at the end of the analysis.
180