Single-Particle Electrodynamics - Assassination Science

σ µν is antisymmetric. The six remaining terms, F i and H i (i = 1, 2, 4), are
further constrained by the requirement of conservation of the fermion current,
(k·J) = 0. Since
u 2 k/u 1 ≡ u 2 b/ 2 u 1 − u 2 b/ 1 u 1 = 0
by the Dirac equation, as does (k·σ·k) by symmetry, this requirement has
no effect on the terms F 1 , F 4 and H 4 . However, since k 2 ≠ 0 in general, the
remaining terms must satisfy
u 2
(
F2 k 2 + H 1 k/γ 5 + H 2 k 2 γ 5
)
u1 = 0.
(E.3)
Now, we can transform the H 1 term by use of the Dirac equation for the
incoming and outgoing fermion states, from which one can verify the identity
u 2 k/γ 5 u 1 ≡ 2m u 2 γ 5 u 1 . For the requirement (E.3) to hold in general, we
therefore require F 2 = 0 and 2mH 1 + k 2 H 2 = 0. Replacing H 1 using this
result, defining a(k 2 ) ≡ H 2 (k 2 )/2m, and replacing the u 2 γ 5 u 1 in the H 2 term
by u 2 k/γ 5 u 1 /2m, the general interaction vertex can then be written
{
(J ·A) = u 2 F1 (k 2 )γ µ + F 4 (k 2 )σ µν k ν + H 4 (k 2 )σ µν k ν γ 5
+ a(k 2 ) ( k/k µ − k 2 γ µ) }
γ 5 u1 A µ . (E.4)
E.3 The classical limit
We now investigate what interaction Lagrangian will be obtained from the
vertex (E.4) in the classical limit.
As we are, in this limit, treating the
fermion as a particle, but the photon as a field, it is appropriate to return
to position space and replace the photon canonical momentum, k, by the
partial derivative of the external potentials and fields, ∂. At the same time,
we must replace the various matrix elements in (E.4) by some sort of classical
counterparts.
desired results:
It is found that the following transformations produce the
u 2 γ µ u 1 −→ U µ + 1
2m ˜Σ µν ∂ ν ,
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