What theories address the fundamental questions about quantum mechanics?

[Fredrik]

Is there a principled, fundamental obstacle to the 'correspondence rules' *ever* being written down, are they, in principle, incapable of being written down?

It is possible in principle to write them down. This is the thread where I realized that. Check out posts #97 and #101. Ignore the quote in #97 that has a list with items numbered from 1-3, and look at the new version of the list in #101 instead. The general idea is: We have to define a hierarchy of theories. Level-1 theories have correspondence rules that we just guessed. Level-(n+1) theories have correspondence rules that can be understood by someone who who understands level-n theories and has access to level-n measuring devices.

Note that theories can't be developed in isolation from each other. A large-n version of classical mechanics may contain, as part of its definition, an instruction manual that tells you how to find some cesium, separate it from its environment, and build a cesium clock. This of course requires knowledge of lower-n versions of both classical and quantum mechanics.

 

[andrebourbaki]

clocks do not perform quantum measurements

In the posts in which Fredrik discusses his projected systematisation of the part of physical theory which is not yet tidy, the correlations between quantum observables and physical measurement devices (and procedures of state preparation as well, I presume), there is a certain amount of discussion of clocks.

It is important to realize that in QM, the Hamiltonian is not an observable, and neither is its conjugate, time. Especially, clocks do not perform quantum measurements and do not reduce the wave packet of anything. The reason for this is physical: they do not amplify anything. Quantum measurement is different from classical measurement precisely in that Geiger counters, photographic emulsions, bubble chambers, etc., all amplify something microscopic to the macroscopic so we can see the pointer, hear the click, see the dot on the photographic plate, see the track of bubbles, etc. Clocks don't do this and that is why $H$ is never treated as an observable.

Of course this does not address the essence of Fredrik's point, but much of it represents philosophy of science more than actual science. I would like to at some point address the essence of Fredrik's project, which is one that many physicists would agree with.

 

[andrebourbaki]

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.

 

[Fredrik]

I don't think a theory of the sort you envisage is possible. The reason is fundamental. You want to be able to derive which objects are to be considered measuring devices corresponding to specific self-adjoint operators, but to have any chance to do that, you must give the measuring devices mathematical definitions. This would make the entire "theory" pure mathematics. The problem is that no piece of mathematics can make predictions about reality on its own. It must be supplemented by non-mathematical statements that tell us how to interpret the mathematics as predictions about results of experiments. So statements of the sort you want to avoid can't be avoided entirely.

 

[andrebourbaki]

Is this a concession that there is an experimentally replicable regularity and no conceivable physical theory can predict it or explain it or even, it seems you go this far, even describe it?

For the rest, your assertions are mostly philosophy, which is not quite the thing to discuss in this forum, I suppose, although I of course am greatly interested in the philosophy of science.

It is not that me and Dirac and Feynman and Bell and Weinberg want to avoid such statements entirely...we are willing to make them at the level of `praxis' like ordering dinner at a restaurant, where we don't use the formalism of physical theory either.
But if the concept of `measurement' is neither defined nor connected by other axioms to the other undefined concepts, as explained in my previous post, then it should not appear in the six fundamental axioms of QM.

 

[andrebourbaki]

Feynmans' opinion about the Axioms of quantum mechanics

`We and our measuring instruments are part of nature and so are, in principle, described by an amplitude function [the wave function] satisfying a deterministic equation [Schrodinger's equation]. Why can we only predict the probability that a given experiment will lead to a definite result? From what does the uncertainty arise? Almost without a doubt it arises from the need to amplify the effects of single atomic events to such a level that they may be readily observed by large systems.

` \dots In what way is only the probability of a future event accessible to us, whereas the certainty of a past event can often apparently be asserted? \dots Obviously, we are again involved in the consequences of the large size of ouselves and of our measuring equipment. The usual separation of observer and observed which is now needed in analyzing measurements in quantum mechanics should not really be necessary, or at least should be even more thoroughly analyzed. What seems to be needed is the statistical mechanics of amplifying apparatus.'

R. Feynman and A. Hibbs, Quantum Mechanics and Path Integrals, New York, 1965, p. 22.

This is quoted and discussed in my The Axiomatisation of Physics, see
http://www.mast.queensu.ca/~jjohnson/HilbertSixth.pdf
and
http://arxiv.org/abs/0705.2554