Single-Particle Electrodynamics - Assassination Science

is over a square region in the x–x ′ plane; the domains of x and x ′ are each
from −ε to ε, independent of the value of the other. If we define the new
variables
x d ≡ x − x ′ ,
x s ≡ x + x ′ ,
the domains of x d and x s are now clearly from −2ε to 2ε. The Jacobian
∂(x, x ′ )
∂(x s , x d ) = 1 2
balances only half of this apparent fourfold increase in the area of integration.
The other half is of course due to the fact that the original square area of
integration has been rotated into a diamond shape. The consequence is that
the limits of integration of the “inner” integral are no longer constant, but
are rather dependent on the value of the variable in the “outer” integral.
Explicitly, if we perform the x d integral outermost, the limits of integration
for the inner x s integral are
|x s | ≤ 2ε − |x d | ;
the domain of integration for x s is now, on the average, half as long as the
naïve maximum-minus-minimum calculation would suggest.
The results of the equivalent considerations in the fully three-dimensional
case are not as obvious—because we are not so adept at visualising six-dimensional
geometry,—but nevertheless proceed in the same manner.
The
self-interactions that we are considering in this chapter are of the general
form (6.4), namely
( )
ab 4 −2 ∫ ∫
4π 3 πε3 d 3 r d 3 r ′ λ(r)f(r, r ′ ).
r≤ε r ′ ≤ε
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