What theories address the fundamental questions about quantum mechanics?

[Fredrik]

I think we have come as far as we can. To discuss this further will not bring us to an agreement about what the best way to handle foundational issues is. But I appreciate that you forced me to clarify some of my points, in particular step 3, because now I understand it better than I did before. So this has definitely been a useful discussion, even though we didn't reach an agreement.

I'm curious about one thing though. What "real issues" would you say that QM has? If QM is a theory only in the sense that it makes predictions about probabilities of possible results of experiments, and those predictions agree with experiments, how can it have issues? I don't think it has any issues other than those that are caused by additional assumptions of the sort that you described as "fantasies".

 

[A. Neumaier]

I'm curious about one thing though. What "real issues" would you say that QM has? If QM is a theory only in the sense that it makes predictions about probabilities of possible results of experiments, and those predictions agree with experiments, how can it have issues? I don't think it has any issues other than those that are caused by additional assumptions of the sort that you described as "fantasies".

The real issue is that the foundations are far from being precise enough to get a consensus about the meaning of QM. My interpretation rule MI is the minimal consensus that _every_ interpretation respects, and it is enough to make many predictions that agree with experiments. But only a small number of experiments that are actually performed fall under that minimal consensus.

The real issue therefore is to augment MI in such a way that it is comprehensive and yet acceptable to everyone.

For currently, beyond the minimal consensus, there are only a number of ad hoc rules employed by experimenters and theorists that force the experiments somehow (and intuitively) into the framework of the theory, and there is where the muddy waters begin. For example, in reality, many quantum observations violate the MI assumption of being independently prepared. But people close the eyes and pretend that they have an ensemble to which the standard Born rule and its consequences can be applied. But this no longer follows from the axioms as stated anywhere I know. And there are many more such issues.

It took me a long time before I began to understand what is going on there, and I am still perfecting my views on this - until I'll be able to write a book that is so clear that people will say: yes, of course, this way QM makes sense.

The discussions here are part of this clarification process: They force me to express myself more clearly than I'd do without the corrective coming from the responses. They indicate to me that I wasn't yet clear enough.

In my mind, QM is far from being the weird theory it is often pictured as. On the contrary, it is a very orderly, intuitive theory in which one can think almost classically if one uses the right visualization. My axioms and interpretation are already far stronger and far less idealizing than those I found in the literature. And I see that some of what I say is already persuasive. As long as the persuasive power of my arguments is still increasing, it makes sense for me to continue such discussions. In the end, quantum mechanics and its interpretation will be as crystal clear as classical Hamiltonian mechanics is.

 

[dextercioby]

[...]It took me a long time before I began to understand what is going on there, and I am still perfecting my views on this - until I'll be able to write a book that is so clear that people will say: yes, of course, this way QM makes sense.
[...]

That's a very ambitious plan you have. I hope you mean to write a book for the somewhat knowledgeables (i.e. for those one which already took a formal/superficial course with all the typical textbook examples which normally lack any mathematical rigorosity or abound in hand-waving arguments) and not for the beginners. That would interest me

In the end, quantum mechanics and its interpretation will be as crystal clear as classical Hamiltonian mechanics is.

That would be really interesting, if made true someday.

 

[A. Neumaier]

That's a very ambitious plan you have. I hope you mean to write a book for the somewhat knowledgeables (i.e. for those one which already took a formal/superficial course with all the typical textbook examples which normally lack any mathematical rigorosity or abound in hand-waving arguments) and not for the beginners. That would interest me

That would be really interesting, if made true someday.

You can get an idea of what I am aiming at if you look at

Arnold Neumaier and Dennis Westra,
Classical and Quantum Mechanics via Lie algebras,
2008.
http://lanl.arxiv.org/abs/0810.1019

It is the draft of an almost finished book (not yet 'the one' I envisioned in my previous mail, but one must work in stages to see what is feasible). Should you or anyone else here read it, I'd appreciate being informed by email (address at my home page) about inaccuracies and suggestions for improvements.

As you'll see, it starts off assuming a little familiarity with physics. But everything used (beyond elementary linear algebra and calculus up to partial derivatives) is actually defined; so a dedicated reader can use it for self-study. The requirements get high only late in the book. Indeed, I know of several 16 year olds who enjoyed reading large fractions of the book. On the other hand, the book already contains some new points of view even for experts.

Chapter 7 contains a first approximation to my thermal interpretation of quantum mechanics. It is much more realistic and down to Earth than anything else I have seen.

My goal for the future foundational book is to have something that nicely explains the world as wee see it, starting from basic axioms (like those presented in post #5 here),
and an exposition of QED (so no nuclear and subnuclear physics). Still missing (compared to the above draft) is most of field theory -- in particular, a rigorous version of QED (the hardest thing, since the literature does not even contain a rigorous definition of what QED should be) and nonequilibrium statistical mechanics (where I have lots of notes but not yet a coherent write-up), which is what will make everything realistic and close to macroscopic physics. In particular, it will be able to describe realistic measurements.

Unfortunately, since I have a full-time job as a math professor, working there on very different topics, work on this is slow. But I make steady progress each time I have a few weeks to concentrate on it.

 

[dextercioby]

I will try to make time to read on your work and will address any possible issues I may find in private, because the guidelines of this forum prohibit us from discussing WIP, but only either published books or published articles in peer-reviewed journals. Your referenced item doesn't currently classify to any of the 2 categories allowed for public debate.

For the record, I think your decision to join the forum made a remarkable increase of quality to the content written here. Also for the record, I remember your name from 2003-2005 when this forum was sharing posts published on http://groups.google.com/group/sci.physics.research/topics. Now the link has gone, but thankfully you joined PF.

On topic now, I remember one objection I made to your set written in post #5, namely not postulating the unique feature of systems of identical particles. If you describe the states by von Neumann density operators, how will this operator 'capture' the symmetrization/antisymmetrization of the tensor product of spaces ? I'm now referring to your statements in post #56 which do not contain a satisfactory answer for me to the questions I raised. Moreover, why shouln't the description of these particular systems be axiomatized ? Can it be then derived from another axioms ?

Thank you

Daniel

P.S. Another side note: if you're a colleague of Prof. Georg Teschl, and as I highly appreciate his work/book on quantum mechanics (it could be viewed as a complement of E. Prugovecki's 1970 book in the sense of providing the Hilbert space solution to the H atom in its simplest quantum mechanical description (I have't seen it in the literature in other place)), it would be nice or convenient for me, if the work you're preparing would have the same mathematical depth as his and as your mentioned book draft.

 

[strangerep]

I will try to make time to read (Arnold's) work and will address any possible issues I may find in private, because the guidelines of this forum prohibit us from discussing WIP, but only either published books or published articles in peer-reviewed journals. Your referenced item doesn't currently classify to any of the 2 categories allowed for public debate.

Although this is correct, under strict interpretation of PF guidelines, I'd be quite
disappointed to be locked out of such discussions as a consequence.

Arnold, it seems to me that such discussions of your public papers which are not
yet published in peer-reviewed places could "legally" take place over in
the Independent Research forum. Indeed, I'm sure such discussion would raise
the overall quality there, as your contributions here have done.

 

[A. Neumaier]

I will try to make time to read on your work and will address any possible issues I may find in private, because the guidelines of this forum prohibit us from discussing WIP, but only either published books or published articles in peer-reviewed journals. Your referenced item doesn't currently classify to any of the 2 categories allowed for public debate.

In view of the comment of strangerep, you might want to open a thread in the Independent Research forum. This must be justified, so you should include some background information such as the one given in your current post. Then that forum would discuss the content matter, while for things such as reporting misprints or making minor suggestions, you should use email.

On topic now, I remember one objection I made to your set written in post #5, namely not postulating the unique feature of systems of identical particles. If you describe the states by von Neumann density operators, how will this operator 'capture' the symmetrization/antisymmetrization of the tensor product of spaces ? I'm now referring to your statements in post #56 which do not contain a satisfactory answer for me to the questions I raised. Moreover, why shouln't the description of these particular systems be axiomatized ? Can it be then derived from another axioms ?

Instances satisfying the axioms are usually not part of the axiom system.
For example, we have the axioms for groups, and then we have - as part of the group theory built on top of that - special constructions such as the groups Sp(n,C), which are instances of a group but of course with additional structure.

Therefore, all parenthetical remarks in my axioms (including the one in A4 mentioning distinguishable particles) are not part of the axiom system but comments for the readers so that they associate the right intuition with the axioms. The list of examples given only has illustrative character and is far from being exhaustive.

I didn't mention indistinguishable particles in my examples for two reasons:
1. One cannot easily specify the set of relevant observables without introducing lots of additional notation or terminology - whereas the explanations of the axioms should be very short.
2. I think that the concept of indistinguishable particles is completely superseded by the concept of a quantum field. The latter gives much better intuition about the meaning of the formalism, and the former (which is difficult to justify and even more difficult to interpret intuitively) is then completely dispensable.

If you are interested in how I think about indistinguishable particles, read Example 5.1.8(iii) on p.99 of the draft of my book, and the discussion of post #25-#41 in the thread https://www.physicsforums.com/showthread.php?t=471125 , as far as it concerns indistinguishable particles. If this doesn't explain enough, please start a new tread with a specific question.

P.S. Another side note: if you're a colleague of Prof. Georg Teschl,

I don't know Georg Teschl, but have a colleague called Gerald Teschl whose office is two doors from mine.

and as I highly appreciate his work/book on quantum mechanics (it could be viewed as a complement of E. Prugovecki's 1970 book in the sense of providing the Hilbert space solution to the H atom in its simplest quantum mechanical description (I have't seen it in the literature in other places),

This is only the tip of an iceberg. Read:
-- Chapter 21 of: BG Wybourne, Classical groups for physicists, Wiley 1974.
-- Section 4.1 and 4.2 of: Thirring, A course in mathematical physics, Vol. III.
-- B. Cordani, The Kepler Problem, Birkh"auser 2003.
-- Barut and Raczka, Theory of group representations and applications, Warszawa 1980.
(The last book has many group-based exercises on the hydrogen atom; probably in Chapter 12 or 13. But I don't have the book here, hence can't check.)

Section 17.5 of my draft book briefly summarizes what's going on. (We were running out of time. The finshed book will have a thorough treatment.)

it would be nice or convenient for me, if the work you're preparing would have the same mathematical depth as his and as your mentioned book draft.

I prefer to express physics in more elementary terms than he does, but the level of rigor should be the same.

 

[A. Neumaier]

the unique feature of systems of identical particles. [...] why shouln't the description of these particular systems be axiomatized ? Can it be then derived from another axioms ?

The most proper way to give an axiomatic approach to the whole of quantum physics is to give a formal definition of the state of the universe, and then to derive everything else from that - since everything we observe is part of the universe, hence must be encoded in its state. I am working towards this goal, but this requires quantum field theory, and as I said, this part of my foundations is far from finished.

 

[dextercioby]

I don't know Georg Teschl, but have a colleague called Gerald Teschl whose office is two doors from mine.

Yes, of course. Sorry, I didn't check his full name.

This is only the tip of an iceberg. Read:
-- Chapter 21 of: BG Wybourne, Classical groups for physicists, Wiley 1974.
-- Section 4.1 and 4.2 of: Thirring, A course in mathematical physics, Vol. III.
-- B. Cordani, The Kepler Problem, Birkh"auser 2003.
-- Barut and Raczka, Theory of group representations and applications, Warszawa 1980.
(The last book has many group-based exercises on the hydrogen atom; probably in Chapter 12 or 13. But I don't have the book here, hence can't check.)

Thank you for the references. The bolded one looks very interesting.

 

[A. Neumaier]

If you describe the states by von Neumann density operators, how will this operator 'capture' the symmetrization/antisymmetrization of the tensor product of spaces ?

The latter is already encoded in the Hilbert space. Any Hermitian, positive semidefinite, linear operator on the N-particle sector of Fock space with trace 1 automatically represents N correctly (anti)symmetrized indistinguishable particles.

 

[A. Neumaier]

I think we have come as far as we can. To discuss this further will not bring us to an agreement about what the best way to handle foundational issues is. But I appreciate that you forced me to clarify some of my points, in particular step 3, because now I understand it better than I did before.

Let me comment your step 3 with a quote from John Bell, taken from Mermin's paper http://arxiv.org/pdf/quant-ph/0612216 :

''Here are some words which . . . have no place in a formulation with any pretension to physical precision: system, apparatus, environment, microscopic, macroscopic, reversible, irreversible, observable, information, measurement. On this list of bad words the worst of all is “measurement”. . . . What exactly qualifies some physical systems to play the role of “measurer”? . . . The word has had such a damaging effect on the discussion, that I think it should now be banned altogether in quantum mechanics.''

But you want to have ''measurement'' figure very prominently in the foundations. You even accept it as self-evident, without needing the slightest explanation:

The only terms that aren't defined by the other steps are "measures" and "represents". The meaning of "measure" is part of what we already know. We don't explain it for the same reason that we don't explain what a function is.

 

[A. Neumaier]

In view of the comment of strangerep, you might want to open a thread in the Independent Research forum. This must be justified, so you should include some background information such as the one given in your current post. Then that forum would discuss the content matter, while for things such as reporting misprints or making minor suggestions, you should use email.

After reading the moderation procedure for the Independent research Forum, it seems better that I'd open this thread. Please let me know (here) whether you have already started to prepare something, and if yes please send me your draft (by email), so that I can build on it.

I didn't mention indistinguishable particles in my examples for two reasons:
1. One cannot easily specify the set of relevant observables without introducing lots of additional notation or terminology - whereas the explanations of the axioms should be very short.
2. I think that the concept of indistinguishable particles is completely superseded by the concept of a quantum field. The latter gives much better intuition about the meaning of the formalism, and the former (which is difficult to justify and even more difficult to interpret intuitively) is then completely dispensable.

If you are interested in how I think about indistinguishable particles, read Example 5.1.8(iii) on p.99 of the draft of my book, and the discussion of post #25-#41 in the thread https://www.physicsforums.com/showthread.php?t=471125 , as far as it concerns indistinguishable particles.

See also posts #54-#60 from https://www.physicsforums.com/showthread.php?t=473423

If this doesn't explain enough, please start a new tread with a specific question.

For the benefit of everyone, I'll start a new thread on indistinguishable particles putting everything together in one place. See
https://www.physicsforums.com/showthread.php?t=474321
https://www.physicsforums.com/showthread.php?t=474293

 

[Fredrik]

Let me comment your step 3 with a quote from John Bell, taken from Mermin's paper http://arxiv.org/pdf/quant-ph/0612216 :

''Here are some words which . . . have no place in a formulation with any pretension to physical precision: system, apparatus, environment, microscopic, macroscopic, reversible, irreversible, observable, information, measurement. On this list of bad words the worst of all is “measurement”. . . . What exactly qualifies some physical systems to play the role of “measurer”? . . . The word has had such a damaging effect on the discussion, that I think it should now be banned altogether in quantum mechanics.''

But you want to have ''measurement'' figure very prominently in the foundations. You even accept it as self-evident, without needing the slightest explanation:

Theories need to be falsifiable. To be falsifiable, they need to make predictions about results of measurements. I don't think the argument needs to be more complicated than that.

Bell is expressing his dissatisfaction with the fact that QM, as defined by a typical list of axioms, looks like a set of rules that tells us how to calculate probabilities of possibilities, instead of like a description of what actually happens (i.e. an interpretation/ontology/illustration/fantasy). When I read his statement, I see what kind of theory he was wishing for, but I don't see a reason to think that such a theory exists.

 

[A. Neumaier]

In view of the comment of strangerep, you might want to open a thread in the Independent Research forum.

After reading the moderation procedure for the Independent research Forum, it seems better that I'd open this thread.

To prepare for this, I decided to put a newer version of the book on the arXiv, but it turned out that to turn my current intermediate version into something reasonably coherent required more work on my part, and I am not yet finished with that. So it will take a bit longer before the (new version of the) book is on the arXiv, ready for discussion.

 

[A. Neumaier]

To prepare for this, I decided to put a newer version of the book on the arXiv, but it turned out that to turn my current intermediate version into something reasonably coherent required more work on my part, and I am not yet finished with that. So it will take a bit longer before the (new version of the) book is on the arXiv, ready for discussion.

A discussion forum for discussing the much expanded version 2 of the book has been approved: https://www.physicsforums.com/showthread.php?t=490492
Please post your comments there.

 

[andrebourbaki]

As you said in post 133,

Theories need to be falsifiable. To be falsifiable they must predict the results of
measurements. (Or predict the observed probabilities of obtaining various results of measurements.) Or, predict the results of experiments, and an experiment must be replicable, and these probabilities are indeed replicable although the individual results are not so replicable. We are, so far, in agreement, and so is John Bell.

But, you do not appreciate Bell's concern: a theory must also be a theory, i.e., precise and unambiguous. His complaint is that the theory, in particular the axioms, do not state what kind of system produces a measurement, and so the relation between 'system' or 'Hamiltonian' and measurement is not clear, but the practicioner is left free to choose
whether to treat a Geiger counter as a quantum system with a Hamiltonian, or as a measurement apparatus, and get two different answers: in the former, after the measurement, the electron is in an entangled state with the Geiger counter, but with the latter, it is in one definite separable state. These two predictions are hard to falsify, but
they are logically contradictory so the usual six axioms don't constitute a 'theory', let alone a falsifiable theory. A theory does not need good taste, skill, etc. to be employed...that is his point, in principle.

 

[Fredrik]

I don't see any of that as a problem. My view is that a theory only needs to assign probabilities to measurement results, given a preparation procedure and a measurement procedure. This means that a definition of a specific theory consists of a purely mathematical part, and a set of correspondence rules that tell us which measuring devices the probability assignments apply to. The usual axioms of QM only define the purely mathematical part of the theory.

 

[andrebourbaki]

If the theory does not tell you which set of rules to apply when, and this is Bell's point, then it is not the theory which is making the predictions, it is the user. That is, QM requires flair, savoir-faire, good taste, it is not a 'theory' in the precise sense of the word.
Bell agreed in print that for all practical purposes, this is not a problem. But from the logical point of view, it is a problem that there is no way to decide what is a measurement device. This problem may become practical very soon, and in two ways. What if one, with nanotechnology, produced a meso-scopic geiger counter? It would then be seen that the measurement axioms were only approximate. The second one is, the measurement part of the theory (observables, probability, reduction of the wave packet) have never been satisfactorily extended to the relativistic regime. What if the two observers attempted to
perform the same measurement at very different speeds and got different results...
Bell disagreed with your comment that he wanted a theory which gives one a picture of reality, he tried to emphasise that that was not his critique. It is this overlap in the applicability of the axioms that was his complaint, and Wigner's too. The theory itself ought to specify precisely when to apply the measurement axioms and reduction and when not to.

 

[Fredrik]

As I said, the way I see it, the theory consists of a purely mathematical part, and a set of correspondence rules. Only the purely mathematical part is covered comprehensively in QM books. The purely mathematical part can't possibly tell you what the measuring devices are. That's the sort of stuff that's covered by the correspondence rules, which unfortunately, aren't covered comprehensively anywhere. I guess that's what you're talking about, but using a slightly different terminology.

I don't think of this as a problem with QM. It's just an annoying but unavoidable feature of science.

 

[andrebourbaki]

We are using a slightly different terminology. We agree that both the math and the 'correspondence rules' are part of the theory. If the 'correspondence rules' are not covered comprehensively *anywhere*, then you are, with different terminology, conceding Bell's point 'the theory does not tell us ...' You also add that for you, this is not a problem. Bell agrees that *for you* this is not a problem, he also agrees that for all practical purposes it is not a problem.

But. Is there a principled, fundamental obstacle to the 'correspondence rules' *ever* being written down, are they, in principle, incapable of being written down? If so, then the theory cannot be written down.

Bell thought that of course the practice of Physics can never be cut and dry, with all procedures written down in advance, he explicitly allowed that the art of finding workable approximations which permit of making practical predictions requires flair, good taste, etc. But I hope you will be fair to his point of view: if even the fundamental theory (six axioms and a few correspondence rules, say for geiger counters and bubble chambers) cannot, even in principle, be stated clearly and correctly in language in a theory, then Fundamental Physics is not theoretical, the theory can never exist, it is therefore illogical not in the sense that it asserts a contradiction or falsity but in the sense that it cannot be expressed logically.

My experience is that half the physicists in the world are Aristotelians and agree with you, and would not be troubled in saying that even fundamental physics evades or transcends logic, or even that there can never be any such thing as fundamental physics, or an exact truth, or a final theory... (this list represents an increasingly radical degree, not everyone would go all the way down the list). But the other half are, like Weinberg and Dirac, Platonists and would agree that Bell's point, if valid, is a defect that hopefully will be fixed eventually.

 

[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 [Schrödinger'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