  We discuss the algebra and the interpretation of the anomalous Zeeman effect
and the spin-orbit coupling within the Dirac theory. Whereas the algebra for
the anomalous Zeeman effect is impeccable and therefore in excellent agreement
with experiment, the physical interpretation of that algebra uses images that
are based on macroscopic intuition but do not correspond to the meaning of this
algebra. The interpretation violates the Lorentz symmetry. We give an
alternative intuitive description of the meaning of this effect, which respects
the symmetry and is exact. It can be summarized by stating that a magnetic
field makes any charged particle spin. We show also that the traditional
discussion about magnetic monopoles confuses two issues, viz. the symmetry of
the Maxwell equations and the quantization of charge. These two issues define
each a different concept of magnetic monopole. They cannot be merged together
into a unique all-encompassing issue. We also generalize the minimal
substitution for a charged particle, and provide some intuition for the
magnetic vector potential. We finally explore the algebra of the spin-orbit
coupling, which turns out to be badly wrong. The traditional theory that is
claimed to reproduce the Thomas half is based on a number of errors. An
error-free application of the Dirac theory cannot account for the Thomas
precession, because it only accounts for the instantaneous local boosts, not
for the rotational component of the Lorentz transformation. This runs contrary
to established beliefs, but Thomas precession can be understood in terms of the
Berry phase on a path through the velocity space of the Lorentz group manifold.
These results clearly reveal the limitations of the prevailing working
philosophy to "shut up and calculate".
