Single-Particle Electrodynamics - Assassination Science

has precessed by an amount ˙σ dt. (That this amount is the same whether
measured in the CACS or MCLF may be recognised, in advance, from the
fact that the Thomas precession relation (2.58) given in the previous section
does not modify the rate ˙σ if the initial velocity v linking the two frames is
zero; corrections are, in this case, in fact second order in dt.) Therefore, we
seek to boost the precessed spin,
(0, σ + ˙σ dt),
by the velocity ˙v dt. We find
[Σ (dt)] = (( ˙v·σ)dt, σ + ˙σ dt) + O(dt 2 ),
and hence
[ Σ ˙ ] = (( ˙v·σ), ˙σ). (2.67)
Now let us connect this MCLF result, (2.67), to the corresponding CACS
result, (2.63): Clearly, the former is identical to the latter, except for the
extra zero-component contribution ( ˙v·σ). To subtract off this quantity, we
note that
[ ˙U] ·Σ ≡ [ ˙U ·Σ ] = −( ˙v·σ),
at the instant t = 0; and U = (1, 0); hence, we subtract ( ˙v ·σ) from the
zero-component by adding the term U[ ˙U ·Σ ]:
( ˙ Σ ) = [ ˙ Σ ] + U[ ˙U ·Σ ]. (2.68)
Now, in constrast to the case of ˙U, the quantity
˙ Σ is actually very important
to our considerations, since it encapsulates the precession of the
three-spin of the particle. Thus, we must be careful to choose the correct
quantity—( Σ ˙ ) or [ Σ ˙ ]—for any mathematical analysis we wish to perform
on such a particle. Usually, we shall find that it is the quantity ( Σ ˙ ) that
we should in fact be considering, because it represents the precession of the
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