To recap, your part 1, from posts 97 and 101,
is the usual axioms of QM (or any theory),
which is mathematics. Your part 2 gives
physical names to some of those maths concepts.
Part 3 is a provisional, subject to improvement, list
of correspondences: to the name of each quantum
observable from part 2, you make correspond a
blueprint for contructing the measurement apparatus,
e.g., a Geiger counter or photomultiplier detector,
plus its instructions on how to use it, how to get
it to interact with the microscopic system, e.g., an
ion or a photon, which is to be measured.
A list of correspondences between QM observables and
construction manuals is not what Dirac would have
called a fundamental theory. A list is not a theory,
even if the list is based on practice and agrees with
experiment; for one thing, because it is not predictive
of something important, which I am going to explain.
In theory, one would want to have some principle which
explained, for many different observables,
$Q_1$, $Q_2$, $Q_3$, \dots, why each corresponding
measurement apparatus, $H_1$, $H_2$, $H_3$, \dots,
was a measurement apparatus for its observable.
Without such a principle, you could not be predictive:
If one cannot, given an observable $Q$, and the Hamiltonian
$H$ of a measurement apparatus, predict whether or not it
measured that observable, then there is something incomplete
or non-fundamental about your theory. Notice that your list cannot do this since it is never complete, it cannot predict `no, this system,
$H'$, will not measure $Q_1$' if $H'$ is not on the list.
(BTW: For theoretical purposes, a system is given when its Hilbert space of quantum states and its Hamiltonian is given. The Hamiltonian could be thought of as the *name* of the system. And the isomorphism class of the Hamiltonian could be thought of as the name of the *kind* of system it is.)
A theory cannot be regarded as fundamental if there is an
experimentally replicable regularity in Nature that the
theory cannot account for, cannot predict. But the real
behaviour of measurement processes, not captured by
the correspondences of your part 3, is such a regularity.
Feynman also thought that although measurement in QM was
pretty much understood, there was a little more that
could be said: what remained to be done is, in his words,
`the statistical mechanics of amplifying devices'.
Without either a) some more axioms connecting Hamiltonians
with observables, or b) some more definitions: of
`measurement' and `observable' that do the same thing,
QM cannot pretend to be a fundamental theory.
For an effort at b), in the spirit of Feynman, see my
http://www.mast.queensu.ca/~jjohnson/ProbQuantMeas.pdf
This has nothing to do with restoring classical
intuitions of `particle' or predicting the result of a
single measurement, for both Nature and Heisenberg have
taught us that the individual `result of a measurement
process' does not have any experimentally replicable
regularity except the probabilistic one, which is
already explained by QM.