Single-Particle Electrodynamics - Assassination Science

we know that the operator equations of motion for expectation values should
correspond to the classical equations of motion (see Section 2.7). But, unlike
most applications of this theorem, in this case the correct formulation
of the classical equations of motion is actually in doubt, whereas the quantum
equation is most definitively known. By analysing the Dirac equation,
we may therefore gain an independent view of the question of the correct
classical limit, as least as far as spin-half particles are concerned.
In Section 4.4.1, we briefly review the physical interpretation of the Foldy–Wouthuysen
transformation of the Dirac equation, and explain why it
is vital in making contact with the classical limit. Then, in Section 4.4.2,
we list the Heisenberg equations of motion arising from the Foldy–Wouthuysen-ed
Dirac Hamiltonian. The subtleties involved in the interpretation of
these equations are then highlighted in Section 4.4.3, and the expressions
compared to those argued for by the author in previous sections.
It should be noted that this section is purely one of review; the author is
satisfied that the existing literature covers the physics admirably; but several
comments are made by the author as to the physical interpretation of the
mathematics involved.
For the purposes of this section only, we use units in which ¯h = 1.
4.4.1 The Foldy–Wouthuysen transformation
In relativistic quantum mechanics, the wavefunction components for particles
and antiparticles are considered together, and indeed may interact with
each other. But there always exist canonical transformations of the wavefunction
(changes of representation) that mix these particle and antiparticle
components together, while still leaving the physical quantities represented
by the theory unchanged, as long as the operators are complementarily transformed.
This means that the components of the wavefunction that appear to
represent antiparticles in one representation will actually be a superposition
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