Single-Particle Electrodynamics - Assassination Science

It can be noted that equation (5.51) includes, as a special case, the
Maxwell–Lorentz theory of macroscopic fields, for which J αβ represents the
“average magnetisation” and “average polarisation” of the particles comprising
the “medium”. However, (5.51) is, as a differential equation, in fact exact
microscopically (within the domain of the classical limit, of course), and does
not require any “averaging” for its validity.
(To make this connection complete, one needs to know that a careful reanalysis
shows that, while we have in the above assumed the dipole moments
to be fixed in magnitude, the equation (5.51) is, in fact, true for any dipole
source densities—whether fixed, or “induced” by external fields. However,
this property lies beyond the requirements of this thesis; and we will, shortly,
introduce assumptions into our analysis that will again restrict the validity
of the results to particles with fixed moments only.)
5.4.4 Solution of the generalised Maxwell equation
We now seek a general solution to (5.51) for arbitrary free charge and dipole
densities, J(x) and ˜J(x). Again, we seek to phrase the problem in such a
way that a minimal amount of new work must be performed. Employing
the four-potential A µ , as we did for the analogous problem in Section 5.3.3,
(5.51) becomes
∂ 2 A ν − ∂ ν (∂ µ A µ ) = J ν + ∂ µ ˜J µν .
In Section 5.3.3, we employed a Lorentz gauge, characterised by the condition
∂ µ A µ LG = 0, to simplify the left hand side; here, that choice yields
∂ 2 A ν LG = J ν + ∂ µ ˜J µν ,
or, on dropping the free electric charge source term J ν (which we have already
treated in Section 5.3),
∂ 2 A ν LG = ∂ µ ˜J µν . (5.52)
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