Single-Particle Electrodynamics - Assassination Science

moment actually cancels with the Dirac moment.) If one now obtains the
Heisenberg equation of motion for the operator p of (4.80), for the Hamiltonian
(4.77), one in fact finds that it now agrees with the accepted force
law of previous sections; the splitting off of the 1/g term disappears. (The
redshift force is not present; but we shall show shortly that it vanishes to the
order of expansion considered here.)
Anandan [11, 12, 14] also came to the same conclusions as Barone, on the
basis of group theoretical considerations, and provided a physical explanation
for the discrepancy between (4.78) and (4.80): essentially, because of the
Thomas precession of the rest frame of the particle, one must perform a further
transformation of H NW to take one into the Fermi–Walker-transported
coördinate system (which the author has termed the “pre-relativistic” coördinate
system in this thesis); in this frame, the relations (4.80) hold rigorously.
The motivation stated by Anandan [11] for this clarification was the statement
by Goldhaber [94] that the coupling of a quantised spin to the Maxwell
field is isomorphic to the interaction of an isospin with the Yang–Mills field;
this is reflected in the coupling (4.80), and of course is destroyed in the naïve
result (4.78).
It may be thought that the transformation of Anandan puts in jeopardy
the original successful application of the Thomas precession effect, namely,
the calculation of the spin-orbit coupling for the energy levels in the hydrogen
atom. However, this is not so: for the hydrogen atom application, the
electron is “moving”, but the frame we are interested in (the rest frame of the
centre of mass of the atom) is not; thus, for the purpose of computing energy
eigenvalues, it is appropriate to use the untransformed Hamiltonian (4.77).
Conversely, one may view Thomas’s original argument for his precession—
the difference between viewing the hydrogen atom in the atom’s frame and
the electron’s co-accelerated frame—as showing why the Anandan transformation
from the former to the latter is necessary for one to obtain relativistically
correct Heisenberg equations of motion for the position operator.
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