Single-Particle Electrodynamics - Assassination Science

6.6.1 Regularisation procedure
Because the generated electromagnetic fields diverge at the worldline of a
pointlike particle, we clearly need to “regularise” the field expressions, i.e.,
modify the problem slightly so that all expressions are finite, and then take
the point limit after we have computed gradients.
One possible approach would be to analyse the fields of the extended
rigid body—as an expansion in ε, at any rate—by integrating the fields
E a n(r d , r s , ε) and B a n(r d , r s , ε) over the three-sphere of integration of r ′ , i.e.,
by performing an integration over the generating volume only. But then we
would be performing no integral over r itself; as a result, a simple integration
in r d –r s space would no longer be possible. Indeed, the author has not been
able to find any tractable way of computing the integrals in this way.
Instead of this, we will instead consider the expressions for a point particle
from the outset, and employ a subtle limiting procedure to obtain our results.
Clearly, we may effectively shrink our body to a point by setting r ′ = 0 in all
expressions, and then examining how the fields behave around the position
r = 0. The quantity ε is now first order in r alone; we shall find that the
resulting expressions do, in fact, possess sufficient orders of ε for us to take
divergences, etc., of the self-fields around the origin of r—the position of the
point particle at t = 0.
There are a number of methods that one can employ to regularise mathematical
expressions around r = 0, but—to the author, at least—the most
conceptually straightforward is to append an extra (Euclidean) dimension
to r-space. In other words, we consider a new, Euclidean four-dimensional
space, referred to as appended space or ˜r-space, possessing explicit Cartesian
position coördinates
˜r ≡ (w, x, y, z). (6.67)
The presence of the extra dimension w generates a continuous range of threedimensional
subspaces r w in ˜r-space; we consider the value of w to be a
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