Is the measurement problem an example of essential incompleteness?

[borman]

Is the measurement problem an example of essential incompleteness?

Two strong remarks have been made about the Quantum Mechanics:
1) it is internally consistent
2) it is complete

One of the mathematicians who were at Princeton when Einstein was there was Kurt Gödel. He was able to show that in any number theory where the axioms were considered complete that there would be theorems that would be true but not derived from the set of axioms. The set of axioms will always be essentially incomplete. To help demonstrate the proof, he used an old unsolved problem since Euclid regarding the inability to derive a formula for prime factorization and inverted it.

So the question that arises when putting the Quantum Mechanics in a formal representation that seeks to ensure that it is internally consistent and complete is: Does essential incompleteness have a place here as well?

If the answer is yes, this would mean there are true theorems that can not be derived from the fundamental axioms. This means there will be observations from experiments that are not solvable from the fundamental axioms of Quantum Mechanics.

One potential candidate could be “the measurement problem”. This is a problem that has not yet been solved intrinsically from the postulates of Quantum Mechanics and may represent an example of something true but not subject to proof from the basic axioms.

The measurement problem deals with the question of how a purely random process tends to give a classical result far more often than it should. Einstein, quite late in his career, brought the question up to Max Born:

“In a letter dated 1954, Albert Einstein wrote to Max Born “Let
phi1 and phi 2 be solutions of the same Schrödinger equation.. . ..When the system is a macrosystem and when phi1 and phi2 are
‘narrow’ with respect to the macrocoordinates, then in by far the
greater number of cases this is no longer true for phi= phi1 + phi 2. Narrowness with respect to macrocoordinates is not only independent of the principles of quantum mechanics, but, moreover, incompatible with them.” (The translation from Born (1969) quoted here is due to Joos (1986).)”

Source is a footnote from:
DECOHERENCE, EINSELECTION,
AND THE QUANTUM ORIGINS OF THE CLASSICAL
Wojciech Hubert Zurek

Abstract: http://arxiv.org/abs/quant-ph/0105127
PDF (60 pages): http://arxiv.org/PS_cache/quant-ph/p.../0105127v3.pdf

[Ivan Viehoff]

So the question that arises when putting the Quantum Mechanics in a formal representation that seeks to ensure that it is internally consistent and complete is: Does essential incompleteness have a place here as well?

If the answer is yes, this would mean there are true theorems that can not be derived from the fundamental axioms. This means there will be observations from experiments that are not solvable from the fundamental axioms of Quantum Mechanics.

Why mention quantum mechanics specifically? Surely there are doubts over the existence of solutions in any such case, eg, classical mechanics.

In particular, there are the Navier-Stokes equations of classical fluid dynamics.

"...mathematicians have not yet proven that in three dimensions solutions {of the Navier-Stokes Equations} always exist (existence), or that if they do exist they do not contain any infinities, singularities or discontinuities (smoothness)." (wikipedia)

One must also be careful by what one means by existence. There is a sense in which the 3-body problem in classical mechanics (the classical motion of 3 bodies moving under gravity) can be shown to have no explicit solution, ie, you can't write it out as a formula in the same way that you can't exhibit a general solution of the quintic by radicals. But obviously the fundamental theorem of algebra tells us that the quintic does have solutions, and we know how to find them. A solution has been proven always to exist in the 3 body case. Although there remains uncertainty about a general existence proof for the n-body problem, that doesn't seem to prevent people carrying out practical numerical simulations of very large n-body models (even with relativistic adjustments), although inevitably a degree of approximation is involved. See http://en.wikipedia.org/wiki/N-body_problem

[Ken G]

I think the idea behind singling out quantum mechanics is that quantum mechanics has more potential to be a complete description of the behavior of particles. In other words, it is where classical mechanics was 200 years ago-- it hasn't met its boundaries of applicability (forgetting for the time being that relativistic quantum mechanics is known to be a bit spotty). I think borman is using Godel to point out that quantum mechanics must have limitations. However, I think the history of science already makes a pretty good case for that!

The second thing borman is wondering about is if the "measurement problem" in quantum mechanics relates to its incompleteness. I think the answer to that is, it's quite possible. Normally, the Godel statement is of a very esoteric and unspecifiable nature-- you don't even know if the Godel statement is true but unprovable, or false and provable. Thus we don't know that the problem would be that quantum mechanics cannot predict certain actual outcomes; it may be that it makes predictions that are demonstrably wrong (that is usually the case with physical theories, again history is pretty clear on that score). So if the Godel statement is in general so nebulous, why should we expect to be able to associate it with something we actually encounter, like the measurement problem?

Well, I don't know, but there is something about the measurement problem that is self-referential and esoteric-- we build physics around a certain understanding of what a measurement is, and there is no guarantee that quantum mechanics respects that understanding. Thus we don't know if we have embedded into quantum axioms a "poison pill" that will give rise to false predictions ultimately, or if our assumptions happen to be true but there are other things that are also true but simply fall outside of science's purview to predict. Perhaps there are truths about quantum systems that simply leave no footprint on what we call objective science. To be quite honest, I suspect that both would be true-- that quantum mechanics makes false predictions in some situations, and there are truths that it has no access to. The standard measurement problem would be more an example of the latter.

[borman]

There is a problem with proving that the integral calculus really does give the correct answer to what a volume is along with other problems where a discrete method is used to approximate a continuum in the vicinity of cusps or other singularities.

The quantum mechanics starts from the premise that discrete packets of energy, hv, form the fundamental axioms. This perhaps allows the representation of the axioms as a number theory where Godel's thoerem can come into play.

The problem that is suggested by the thread opening is that it appears that the measurement problem can not be resolved intrinsically by derivation from the fundamental axioms. It is apparent that after a large number of measurements we appear to get a classical result, but there is no derivation, intrinsic to the axioms, that this should always result. One should from time to time get some unusual random large scale results, but these predictions are never observed with the frequency that they are predicted to occur.

One might think there were "hidden variables" intrinsic to quantum mechanics that are governing this process. But quantum mechanics is complete. If you add or take away anything from it, then the 11 decimal accuracy begins to go away.

Possible solutions to this problem motivates my interest in non-unified theory. As a possible way around the Godel problem posed by a non-unified two brane approach is the phrase: "Whatever I do not know, my brother knows". If one supposes that the two branes are unified, then one just has a larger set of axioms which then falls victim to the Godel wrench. But if they can complement but remain separate from each other they can address each other's inadequacies as a complete theory as regards the Godel issue.

But they can still influence each other in ways not to be found intrinsic to each's axioms. Suppose there is something like a classical brane that is the foundation for General Relativity. This brane tells the Quantum Mechanics Brane to give the classical results that is at the heart of the measurement problem. But then the non-local nature that occasionally shows itself in the quantum mechanical brane is ill defined when it comes to metric tensors trying to interpret it. These would represent the anomalous forces from the perspective of the GR brane that could be the source of the Dark Sector as well as some of the other locally seen anomalies to GR.

There was a paper by Steven Boughn that rather looked at a one-brane but two toned version that had a rather heretical notion that the quantum brane goes away leaving the classical brane in its place. As a possible result, it was suggested that the non-local states do not contribute to gravitational mass. The idea was that superfluids still represent a mixed state.

By way of contrast, the confusion to the metric tensor of the non-local may be the source of additional acceleration not normally indigenous to the GR brane. There has been some experimental evidence that supports this notion where the stirring of superfluids have induced an acceleration in gyroscopes that is many orders of magnitude greater than what is allowed by classical frame-dragging.

[Disinfo Agent]

Is the measurement problem an example of essential incompleteness?

Two strong remarks have been made about the Quantum Mechanics:
1) it is internally consistent
2) it is complete

One of the mathematicians who were at Princeton when Einstein was there was Kurt Gödel. He was able to show that in any number theory where the axioms were considered complete that there would be theorems that would be true but not derived from the set of axioms. The set of axioms will always be essentially incomplete. To help demonstrate the proof, he used an old unsolved problem since Euclid regarding the inability to derive a formula for prime factorization and inverted it.

So the question that arises when putting the Quantum Mechanics in a formal representation that seeks to ensure that it is internally consistent and complete is: Does essential incompleteness have a place here as well?

In the natural sciences, you can always look for observational data to decide whether a proposition is true or not. You can never prove anything absolutely -- but on the flip side you generally have independent evidence (not derived from a set of theoretical postulates) to make a case for or against a claim.

[Disinfo Agent]

On the other hand...

'[...] call off the search for a theory of everything. Physicist David Wolpert, in an article published in the prestigious Physica D (vol. 237, pp. 1257–1281, 2008), has shown that -- at best -- we can achieve a theory of almost everything. Wolpert’s work is very technical, but its implications are spectacular. Unlike the above mentioned limits to knowledge, which come out of empirical disciplines, Wolpert used logic to prove his point, following in the steps of the famous incompleteness theorem demonstrated by Kurt Godel in 1931. (An accessible summary of Wolpert’s discovery can be found in an article by P.-M. Binder in Nature, 16 October 2008.)' "No Theory Of Everything Is Possible", By Massimo Pigliucci @ Scientific Blogging

[Ken G]

From that article:

To name a few examples, relativity theory imposes limits to how fast information can be transferred (the speed of light); chaos theory tells us that the behavior of complex non-linear systems cannot be predicted after a few time steps, despite the fact that these systems are deterministic; quantum mechanics says that we cannot measure all the properties of a particle at the same time (Heisenberg’s principle); and complex systems theory has established the principle of intractability, which shows that the behavior of some physical systems cannot be predicted before actual observation of such systems.

Note that only one of these examples is from quantum mechanics-- showing that we already had plenty of difficulties in knowing completely our reality even in classical physics.

[borman]

Some references related to the thread

The David Wolpert paper from the arxiv prior to publishing:
Physical limits of inference
http://arxiv.org/abs/0708.1362

On comparing the Halting problem to the Gödel theorem:

The Halting Problem, and Gödel's Theorem
http://www.quadibloc.com/math/undint.htm

On Wigner’s friend: http://en.wikipedia.org/wiki/Wigner%27s_friend

An Adler paper distinguishing decoherence from the measurement problem:
Why Decoherence has not Solved the Measurement Problem: A Response to P. W. Anderson
http://arxiv.org/abs/quant-ph/0112095


Of tangential interest:

Is the black hole area theorem always classically valid?
By Eling and Beckenstein
http://arxiv.org/abs/0810.5255

A couple loopholes are left to be closed before a classical conclusion is forced, but it remains a candidate.

IMHO, in the event the loopholes are closed, it may be relevant to distinguish between what is needed for classical validity and a throwback to pre-Planck classical ideas. The neo-classical criteria may not be as familiar and may appear as strange as the quantum world.

On the relationship between instability and Lyapunov times for the 3-body problem
http://arxiv.org/abs/0810.5461

This last is brought up in connection with the early arrival times of a number of long period commets.