Single-Particle Electrodynamics - Assassination Science

tion. Furthermore, if the spin-half particle does not actually come in contact
with any other physical electromagnetic sources, then there are two further
simplifications: the anapole moment has no effect whatsoever; and, in the
classical limit, the only parts of the structure functions that have any effect
are the static values of the moments: q, d and µ.
In the remaining sections of this appendix, we provide brief review of the
proofs of these properties, for convenience; there are no new results herein.
E.2 Quantum field theory
Viewed from the perspective of quantum field theory, the electromagnetic
interaction of a spin-half particle with an external field is fundamentally
a vertex of the fermion with an external photon. Seeing as the photon is
fundamentally described by the four-potential, A(x), any coupling to it in
the Lagrangian must be of the general form (J ·A), where J might include
operators, such as ∂, when we are viewing the process in position space.
Let the fermion’s initial canonical four-momentum (four-wavevector) be
b 1 , and its final canonical four-momentum b 2 . It is convenient to replace
these two quantities by the photon’s canonical four-momentum, k (which by
canonical momentum conservation is equal to b 2 − b 1 ), and the sum of the
fermion canonical momenta, B ≡ b 1 + b 2 . Together with the various matrix
quantities characterising the Dirac algebra—namely, 1, γ 5 , γ µ , γ 5 γ µ , σ µν and
γ 5 σ µν —we can proceed to construct the most general coupling to the photon
that is possible. Writing out all possible terms blindly, apart from those that
are obviously dependent, we have
〈2|J µ (k)|1〉 = δ(b 2 − b 1 − k) u 2 (F µ + G µ + H µ + K µ ) u 1 ,
(E.1)
where
F µ ≡ F 1 γ µ + F 2 k µ + F 3 B µ + F 4 σ µν k ν + F 5 σ µν B ν ,
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