Single-Particle Electrodynamics - Assassination Science

Noting now the definition
we thus find
F ≡ ∂∧A,
ṗ = qF ·U, (2.80)
the covariant expression of the Lorentz force law.
Now, to apply the above manifestly covariant Lagrangian formalism to
particles possessing “internal” degrees of freedom—which we shall describe
using the normalised spin four-vector Σ ,—one finds that one has to deal with
various new subtleties that are introduced. Firstly, for the Euler–Lagrange
equations of motion arising from the translational degrees of freedom z α , we
shall find that the normalised four-spin Σ will appear among the factors that
need to be subjected to the proper-time derivative operator d τ . The question
then arises as to whether we should use the partial or covariant proper-time
derivative of this quantity. The author confesses that his viewpoint on this
subject has changed quite recently: he [65] originally believed that it should
be the covariant derivative; however, one needs to note that the Lorentz
dot-product,
d τ (A·B),
must be treated consistently; if one differentiates the components of (say) A,
then one must of course also differentiate the components of B. This implies
a partial derivative. (This requirement will be used extensively in Chapter 5.)
This point is subtle, for the case of the dipole force law, because the “other
quantity” is in fact the electromagnetic field F µν , for which we must in fact
use the convective derivative: a third version of the proper-time derivative.
This issue was, in fact, only discovered by the author a few days before the
printing of this thesis; it is discussed in more depth in Sections 4.2.1 and
4.3.4.
The second issue arising in connection with the use of the four-spin is
the question of obtaining the rotational equations of motion for the system,
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