Single-Particle Electrodynamics - Assassination Science

We define
η 0 ≡ 1 ( ) 4 −2 ∫ ∫
4π 3 πε3 d 3 r d 3 r ′
r≤ε r ′ ≤ε
≡ 1
4π .
The six-dimensional region of integration in r–r ′
constraints
space is defined by the
|r| ≤ ε,
|r ′ | ≤ ε. (6.13)
The transition to the variables r d and r s mixes components of r and r ′ . Since
we know, from the discussion of the previous section, that the three-vector r d
will dominate our considerations, with r s playing a considerably lesser rôle,
we shall usually choose to perform the the r d integral outermost. The region
of integration of r d will then be determined only by the maximum values
attainable by r d ; but the region of integration for r s will depend on the value
of r d in the outer integral. It is clear, from the definitions (6.11), that r d
may take any value within a three-sphere of radius 2ε, i.e.,
r d ≤ 2ε, (6.14)
and thus this larger three-sphere is the region of integration for r d ; we shall
call this volume V d . The constraints (6.13) require that
|r s + r d | ≤ 2ε, (6.15)
|r s − r d | ≤ 2ε; (6.16)
i.e., the region of integration is the intersection of these two regions. Now, for
any given value of r d , these constraints define two three-spheres in r s -space,
each of radius 2ε, that are offset by ±r d from the origin; their common
volume—which looks like a three-sphere with a central slice taken out and
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