Single-Particle Electrodynamics - Assassination Science

that fall off as R −1 ), then we are ignoring the mechanical energy contained
in the near fields (those that fall off as R −2 )—and, indeed, the interference
between the near and far fields. Of course, if the system is accelerated for a
certain finite period of time, and then left alone forever after, the near fields
will eventually stabilise themselves, and only the mechanical energy pilfered
by the radiation fields will come off the bottom line of the balance sheet
for the mechanical energy. But this small amount of information is rarely
acceptable to us: we would generally like to know what is happening now to
the electric charge—as it is radiating—not what it will be doing next week.
Thus, we clearly need to expand our considerations, to also take into account
the near fields of the charge.
But now we hit another problem: as we approach the position of the
point charge, we find that the field (and hence mechanical energy density)
expressions grow divergently. We must therefore be extremely careful how
we evaluate integrals of the mechanical field energy density: we need to
“regularise” them in such a way that we avoid any danger of manipulating
infinite expressions meaninglessly. One way we can do this is to let the point
particle have a finite size to start off with, and then shrink it to a point at the
end of the calculations. Another way is to exclude a small region surrounding
the charge, and take into account the energy and momentum crossing the
boundary of this region when computing our balance sheets. Whichever way
we proceed, we find that we can, in fact, obtain meaningful answers.
But there is another, more deeply philosophical question that one may
ask: who is responsible for this radiation reaction force? We have figured out
what its effects are—by some nifty detective work,—but we have not really
pinpointed its fundamental source. The Lorentz force law—which we can derive,
most elegantly, from the interaction Lagrangian—does not give us any
clue whatsoever that such a force is lurking on our doorsteps. Where else,
then, could it come from? Maybe we have to add extra terms to our fundamental
Lagrangian, by hand, that yield the correct force? Unfortunately—or,
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