Single-Particle Electrodynamics - Assassination Science

u 2 σ µν u 1 −→ ˜Σ µν ,
u 2 σ µν γ 5 u 1 ≡ u 2 ε µναβ σ αβ u 1 −→ ε µναβ ˜Σαβ ,
u 2 γ µ γ 5 u 1 −→ Σ µ ,
where ˜Σ is the classical unit spin tensor, and Σ the corresponding spin vector.
(The last term in the first expression takes into account the generation of
the “Dirac” magnetic moment in the Foldy–Wouthuysen representation, for
a particle that has purely an electric charge in the Dirac representation [88].)
Since k → ∂, the structure functions F 1 (k 2 ), F 4 (k 2 ), H 4 (k 2 ) and a(k 2 ) can
be interpreted as functions of the d’Alembertian operator, ∂ 2 ≡ (∂ ·∂). We
assume that these structure functions are analytic everywhere, so that they
can be expanded as a power series in ∂ 2 ; for example
a(∂ 2 ) ≡ a 0 + a 1 ∂ 2 + a 2 ∂ 4 + . . . .
We thus find that the most general interaction Lagrangian, in the classical
limit, for spin-half particles is given by
L int = F 1 (∂ 2 )(U ·A) + 1 2
{
F1 (∂ 2 }
)
2m + F 4(∂ 2 ) (· ˜Σ ·F ·)
− H 4 (∂ 2 )(· ˜Σ · ˜F ·) − a(∂ 2 )(Σ ·J ext ),
where we have used the inhomogeneous Maxwell equation,
(E.5)
∂ 2 A − ∂(∂·A) = J ext
to express the a(∂ 2 ) term in terms of the “external” current J ext (x) generating
the potential A(x), and where ˜F is the dual electromagnetic field tensor.
Already, one can see in (E.5) the familiar interactions of the classical
limit: electric charge; magnetic dipole moment—with its “pure Dirac” and
anomalous terms; and electric dipole moment. The final interaction term,
392