Single-Particle Electrodynamics - Assassination Science

ut in which they are pure, pointlike, structureless electric charges; on the
other hand, they are classical objects, in the Newton–Wigner representation,
in which their operators act quite in accord with classical mechanics, but
in which their electromagnetic moments are more complicated: they still
have electric charge; but through the Foldy–Wouthuysen transformation they
acquire a magnetic moment, and (less well-known) an electric charge radius
(manifested in the “Darwin term” in the Hamiltonian; see [89, 90, 91, 92]).
(In fact, if one wishes to be rigorous, the Foldy–Wouthuysen transformation
of a pure point charge in the Dirac–Pauli representation induces more
than just a magnetic moment and an electric charge radius; these are just
the lowest-order effects. The Sachs form factors [181, 182] are defined precisely
so that one may relate the properties of the “Newton–Wigner face”
of a fermion—real classical particles measured in real experiments—to the
“Dirac–Pauli face” of the fermion—which, in the case of the massive leptons,
is a structureless point charge. However, it should be noted that the Sachs
form factors effectively only take into account the lowest order effects of the
Foldy–Wouthuysen transformation, namely, those listed above; despite their
usefulness, the physical validity of these form factors—in terms of the above
interpretation—cannot be assumed away from the region around q 2 = 0.
See [92] for a thorough discussion of the issues involved.)
4.4.2 The Heisenberg equations of motion
Let us start with the minimally-coupled Hamiltonian in the Dirac–Pauli representation
for a pure electric charge q:
H DP = βm + α·(b − qA). (4.74)
Since we know that the Foldy–Wouthuysen transformation will yield a “pure
Dirac” magnetic moment,
µ Dirac = q σ, (4.75)
2m
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