An argument against Bohmian mechanics?

[FeynmanFtw]

There is a paper by Arnold Neumaier, where it is argued that Bohmian mechanics, is simply wrong, because it doesn't predict all the results that we observe from experiment. See here.

Neumaier wrote down his argument for a particle in the ground state of a harmonic oscillator, but there's nothing fundamental about this choice. It was there to frame the argument at its simplest and clearest. If, instead, we chose a linear combination of the ground and (say) first excited states, the answers obtained with quantum mechanics and Bohmian mechanics would once again disagree, and because there's a difference in energy between the ground state and first excited state there would be a relative phase that would survive the step of taking the expectation value. Bohmian mechanics would no longer predict that the particle would sit stationary in the same spot -- it would undergo some evolution. But it still wouldn't give the right answer for the relative phase: the imaginary part of the correlator is generically nonzero in quantum mechanics (the correlator is not Hermitian so it is not guaranteed a real spectrum), but always zero in Bohmian mechanics.

What are your thoughts?

 

[StevieTNZ]

There is a paper by Arnold Neumaier, where it is argued that Bohmian mechanics, is simply wrong, because it doesn't predict all the results that we observe from experiment.
...
What are your thoughts?

You may not know this, but Arnold Neumaier (https://www.physicsforums.com/members/a-neumaier.293806/) is a science advisor on this forum, so I'll tag him (@A. Neumaier) so you may get an answer from the man himself.

 

[Demystifier]

I have seen (and studied) a lot of papers who claim that BM contradicts observations and/or standard QM. About 99% of these papers make the same mistake, by ignoring the Bohmian quantum theory of the measurement process. The paper by Neumaier is not an exception. When the theory of quantum measurements is taken into account, it turns out that non-relativistic BM and standard non-relativistic QM make the same measurable predictions in all possible cases.

Arguing that BM contradicts QM and ignoring the quantum theory of measurement is like arguing that perpetuum mobile is possible and ignoring the energy-conservation law. No matter how detailed and clever your proposal is, it fails due to a very simple general theorem that you didn't take into account.

 

[Demystifier]

Given the general objections above, let me also add that for the specific proposal by Neumaier the following recent paper is also relevant
https://arxiv.org/abs/1610.03161
especially Sec. 3.2. This papers shows, in a different context, what goes wrong when time-correlations in QM are interpreted naively, without taking into account the effect of measurement.

 

[A. Neumaier]

When the theory of quantum measurements is taken into account, it turns out that non-relativistic BM and standard non-relativistic QM make the same measurable predictions in all possible cases.

In a universe consisting of a single hydrogen atom there is no observer who could perturb the electron's position, so that the theory of quantum measurements could apply.

Moreover, in all possible cases, the Bohmian theory of quantum measurements takes an average over all possible universes with probabilities determined from a preordained distribution that has no rational explanation (except that it leads to the desired Born law). However, in reality we have only one universe (at least all measurements are made in the same universe), and one needs an explanation why repeated measurements made in this unique universe in time should have the same distribution as when one takes one measurement each in each of the universes according to this preordained distribution. One would need some sort of ergodic theorem to ensure that. but such a theorem has never be proved. In view of counterexamples of toy universes such as the hydrogen atom it seems unlikely that such an ergodic theorem is valid in a reasonable generality.

Thus the Bohmian theory of quantum measurements is incomplete in a fundamental respect, and therefore, in its present state, fails to account for the quantum behavior in the single universe we know of.

 

[Demystifier]

One would need some sort of ergodic theorem to ensure that.

Neither classical statistical mechanics, nor quantum statistical mechanics, nor Bohmian mechanics, really need ergodic theorem to work properly. But that's another topic.

 

[A. Neumaier]

Neither classical statistical mechanics, nor quantum statistical mechanics, nor Bohmian mechanics, really need ergodic theorem to work properly.

Well, prove your claim by meeting my objection!

 

[Demystifier]

Well, prove your claim by meeting my objection!

We would need first to dwell deeply into foundations of statistical mechanics, which, as I said, is another topic.

 

[A. Neumaier]

We would need first to dwell deeply into foundations of statistical mechanics, which, as I said, is another topic.

Then point to a research article where your claim is proved. Or, if none exists, write one that reveals to us the secret of this great news. Or, if you can't do this, refrain from making such a revolutionary claim!

 

[Demystifier]

Then point to a research article where your claim is proved. Or, if none exists, write one that reveals to us the secret of this great news. Or, if you can't do this, refrain from making such a revolutionary claim!

As I said, that's another topic. If you want, you can create a new thread on it, where I will be happy to say more.

 

[A. Neumaier]

You made this extraordinary claim here, so prove it here!

This is a thread about an argument against Bohmian mechanics, and the hydrogen example of the leading post is a counterexample to the ergodic behavior of a Bohmian universe. So refuting the argument requires to show why no ergodic theorem is needed!

 

[Demystifier]

... to show why no ergodic theorem is needed!

Statistical physics without ergodicity:

https://www.jstor.org/stable/1215826?seq=1#page_scan_tab_contents

https://arxiv.org/abs/1103.4003

https://arxiv.org/pdf/cond-mat/0105242v1.pdf

Bohmian version of the H-theorem:

http://www.sciencedirect.com/science/article/pii/037596019190116P

 

[Demystifier]

In view of counterexamples of toy universes such as the hydrogen atom it seems unlikely that such an ergodic theorem is valid in a reasonable generality.

Quite generally, very simple systems with a small number of degrees of freedom (and hydrogen atom is certainly an example of such a system) are usually bad examples for understanding how and why statistical physics work.

 

[A. Neumaier]

Quite generally, very simple systems with a small number of degrees of freedom (and hydrogen atom is certainly an example of such a system) are usually bad examples for understanding how and why statistical physics work.

But they serve as counterexamples to sweeping statements that need proper assumptions to be valid. Thanks for your references.

 

[Demystifier]

what goes wrong when time-correlations in QM are interpreted naively, without taking into account the effect of measurement.

It just ocured to me that it can be generalized in a very simple and illuminating way. Let ##A## and ##B## be two quantum observables. Their correlator is
$$\langle \psi |AB| \psi\rangle ...(1)$$
Can this correlator be measured? By measuring the correlator, one usually means measuring ##A## and ##B## separately and then combining the measurement results. However, if
$$[A,B]\neq 0$$
then measurement of ##AB## is not equivalent to measurement of ##A## and ##B## and subsequent multiplication of the measurement results. In this sense, by measuring ##A## and ##B## separately, one cannot measure the correlator (1).

As a special case, one can take ##A=x(t_1)##, ##B=x(t_2)##. If
$$[x(t_1),x(t_2)]\neq 0$$
then separate measurements of ##x(t_1)## and ##x(t_2)## will not give the time correlator
$$\langle \psi |x(t_1)x(t_2)| \psi\rangle$$

 

[A. Neumaier]

Statistical physics without ergodicity:

https://www.jstor.org/stable/1215826?seq=1#page_scan_tab_contents

https://arxiv.org/abs/1103.4003

https://arxiv.org/pdf/cond-mat/0105242v1.pdf

Bohmian version of the H-theorem:

http://www.sciencedirect.com/science/article/pii/037596019190116P

These papers just replace ergodicity by something similar, and also testify to the fact that the results are far from consensual. In any case one needs to prove something that relates certain averages in time with the ensemble averages, and the authors need to make make more or less plausible additional assumptions (molecular chaos, statistical independence, macroscopic observables) to obtain such results. In particular, nothing is said about the properties of microscopic observables. But Bohmian mechanics claims to predict correctly microscopic observations, hence must be able to derive these for measurements on the single universe!

The final, Bohmian paper is wishful thinking. The author says after (16): ''so that S is bounded from above (by zero). This fact, together with the fact that S cannot decrease, will be regarded here as proof that S eventually approaches its maximum value, i.e. S##\to##0. [...] Thus, provided one performs measurements of a sufficiently coarse accuracy with respect to configuration-space volumes, one will see a distribution'' as required for the uncertainty relation to hold (as was argued earlier). It is of course only a ''proof'' with a glaring gap! In the analogy to the Boltzmann H-theorem that the author employs, claiming what the author claims is equivalent to the claim that, because of the H-theorem, each dilute gas, provided one performs measurements of a sufficiently coarse accuracy, is in equilibrium. But the latter is obviously not the case.

Moreover, even if the correct distribution would have been derived, it is only a distribution for an ensemble of universes and not a distribution related to actual measurements in a single universe.

 

[Demystifier]

I think our disagreement is not so much about Bohmian mechanics, but about the general concept of probability. This can be illustrated by the following example. Take one letter from the set {A,B}, and don't tell me which one you took. I claim that, from my perspective, the probability that you took A is p(A)=0.5. And you don't agree that I can assign probability in this way. As long as we cannot agree on such a basic thing, there is no point in arguing about more complex problems such as probability in Bohmian mechanics.

 

[rubi]

I think our disagreement is not so much about Bohmian mechanics, but about the general concept of probability. This can be illustrated by the following example. Take one letter from the set {A,B}, and don't tell me which one you took. I claim that, from my perspective, the probability that you took A is p(A)=0.5. And you don't agree that I can assign probability in this way. As long as we cannot agree on such a basic thing, there is no point in arguing about more complex problems such as probability in Bohmian mechanics.

You can of course assign that probability, but if the outcome of the experiment is determined by a deterministic theory, then your assignment will very likely not agree with the experiment, unless you can prove that the system evolves into the state A half of the time. In a deterministic theory, probabilities are objectively determined by the theory. Any "best guess" of the probabilities on the basis of a MaxEnt principle will be wrong unless you can prove that the probabilities that are objectively determined by the theory agree with the MaxEnt best guess. Ergodicity is of course just one of many mechanisms for such a proof, but Arnold is right that you cannot escape the necessity of a proof. If the theory predicts that the system will evolve into state A with certainty, independent of the initial conditions, then your p(A)=0.5 assignment will be way off.

 

[Demystifier]

You can of course assign that probability, but if the outcome of the experiment is determined by a deterministic theory, then your assignment will very likely not agree with the experiment, unless you can prove that the system evolves into the state A half of the time. In a deterministic theory, probabilities are objectively determined by the theory. Any "best guess" of the probabilities on the basis of a MaxEnt principle will be wrong unless you can prove that the probabilities that are objectively determined by the theory agree with the MaxEnt best guess. Ergodicity is of course just one of many mechanisms for such a proof, but Arnold is right that you cannot escape the necessity of a proof. If the theory predicts that the system will evolve into state A with certainty, independent of the initial conditions, then your p(A)=0.5 assignment will be way off.

I don't understand your philosophy. Just because you cannot prove that something is true doesn't mean that it isn't true. If I assign probability in Bohmian mechanics in some heuristic way by a kind of maxent principle, and if that assignment leads to predictions which agree with experiments, then I have evidence that my heuristic approach works well, even if I don't have a "proof". Physics is not mathematics. Mathematics seeks proofs, but physics seeks evidence.

 

[rubi]

I don't understand your philosophy. Just because you cannot prove that something is true doesn't mean that it isn't true. If I assign probability in Bohmian mechanics in some heuristic way by a kind of maxent principle, and if that assignment leads to predictions which agree with experiments, then I have evidence that my heuristic approach works well, even if I don't have a "proof". Physics is not mathematics. Mathematics seeks proofs, but physics seeks evidence.

Well, if something is experimentally true, but a mathematical proof is lacking, then this doesn't mean that the theory is wrong. It's just an open problem of the theory then. This is of course possible. However, I would say that it is a critical open problem in the case of Bohmian mechanics, if such a proof cannot be given, since there are so many interpretations of QM that make the same predictions, so we can't exclude them on the basis of experiments. In such a situation, mathematical consistency is the only objective way to exlude some interpretations. Without a criterion such as mathematical consistency, only personal preference remains.

However, if I understand Arnold correctly, he says that he has an example that contradicts the claims of BM, so the situation might be worse than just the lack of a proof. I haven't read the paper yet, so I won't take sides yet.

 

[Demystifier]

However, if I understand Arnold correctly, he says that he has an example that contradicts the claims of BM

I am sure that he has not such an example, even if he thinks he has.

Sure, not everything is rigorously proved in Bohmian mechanics, but as some of the papers I mentioned above show, not everything is rigorously proved even in classical statistical mechanics. Nevertheless, not many physicists complain about foundations of classical statistical mechanics. Indeed, Bohmian mechanics is very much similar to classical statistical mechanics, so it is not reasonable (and not fair) to expect that Bohmian mechanics should be formulated more rigorously than classical statistical mechanics.

 

[rubi]

Sure, not everything is rigorously proved in Bohmian mechanics, but as some of the papers I mentioned above show, not everything is rigorously proved even in classical statistical mechanics. Nevertheless, not many physicists complain about foundations of classical statistical mechanics.

But in the case of classical mechanics, we can test the underlying theory directly. The tests don't rely on statistical predictions. We can throw stones and measure their trajectory to test Newtonian mechanics. Hence, we have very good evidence that Newtonian mechanics is (approximately) correct and thus, most people are satisfied with the heuristic arguments that lead to statistical mechanics. More rigorous proofs are of course still desirable.

Indeed, Bohmian mechanics is very much similar to classical statistical mechanics, so it is not reasonable (and not fair) to expect that Bohmian mechanics should be formulated more rigorously than classical statistical mechanics.

But the situation is different in Bohmian mechanics. While we can convince ourselves of the correctness of Newtonian mechanics directly, any test of Bohmian mechanics relies critically on the correctness of its statistical predictions, so a correct derivation of the probabilities is a critical element of Bohmian mechanics. Given the fact that there is no other possibility to falsify the theory other than by its statistical predictions, I think it's fair to expect Bohmians to investigate this issue much more thoroughly, especially since BM makes claims that many people find quite extraordinary.

 

[Demystifier]

But in the case of classical mechanics, we can test the underlying theory directly. The tests don't rely on statistical predictions. We can throw stones and measure their trajectory to test Newtonian mechanics. Hence, we have very good evidence that Newtonian mechanics is (approximately) correct and thus, most people are satisfied with the heuristic arguments that lead to statistical mechanics. More rigorous proofs are of course still desirable.But the situation is different in Bohmian mechanics. While we can convince ourselves of the correctness of Newtonian mechanics directly, any test of Bohmian mechanics relies critically on the correctness of its statistical predictions, so a correct derivation of the probabilities is a critical element of Bohmian mechanics. Given the fact that there is no other possibility to falsify the theory other than by its statistical predictions, I think it's fair to expect Bohmians to investigate this issue much more thoroughly, especially since BM makes claims that many people find quite extraordinary.

Well, without a direct experimental evidence for Bohmian mechanics similar to those for Newtonian mechanics, I don't think that more rigorous proofs of the statistical aspects of Bohmian mechanics would contribute much to the general acceptance of Bohmian mechanics.

After all, when Mach criticized Boltzmann for his statistical theory of atoms, Mach didn't complain about Boltzmann's non-rigorous statistical arguments. He complained that there is no direct evidence for individual atoms as such. Loosely speaking, Bohmian mechanics today is what Boltzmann theory was at the end of 19th century.

 

[rubi]

Well, without a direct experimental evidence for Bohmian mechanics similar to those for Newtonian mechanics, I don't think that more rigorous proofs of the statistical aspects of Bohmian mechanics would contribute much to the general acceptance of Bohmian mechanics.

Well, before anyone can accept BM as the "correct" interpretation of QM, it must first be clear that BM is an actual interpretation of QM in the first place. Bohmians often claim that there was a general theorem that BM predictions agree with QM predictions, but apparently such a theorem has never been rigorously established, so we don't know for sure whether BM is indeed an interpretation of QM. Recently, I have seen some papers pop up that claim to prove a negative result (Arnold's paper is only one of them) and big names like Hagen Kleinert appear among the authors. Unless a proof exists that BM reproduces QM, I don't think that BM can be considered to be on par with the other interpretations of QM.

 

[stevendaryl]

Here's my complaint or confusion about Bohmian mechanics, which is the role of the wave function. It's not just a description of our knowledge in Bohmian mechanics, but it is a real thing, having real effects. As was mentioned in A. Neumaier's paper, for a bound state, the Bohmian value for velocity, [itex]\vec{v} = \frac{\hbar}{m}Im(log(\psi))[/itex], vanishes, since the wave function is real. So in the Bohmian picture, a hydrogen atom consists of a proton with an electron sitting at a fixed location above the proton, held in place by the balance between the Coulomb force and the "quantum force" (the effective force corresponding to the "quantum potential" [itex]V_Q = \frac{-\hbar^2}{2m} \frac{\nabla^2 |\psi|}{|\psi|}[/itex]). So you need to know the wave function in addition to knowing the positions of all the particles.

I guess that's not very different from classical field theory, where you have to solve Maxwell's equations for the field in conjunction with solving the equations of motion of the particle in the presence of that field. But there are a couple of striking differences:

1. The wave function is a field in configuration space, not physical space.
2. Solving the wave function for a single particle doesn't involve the location of the particle. So particles don't affect the wave function at all. So it's a philosophical violation of the physics rule of thumb (generalizing from Newton's third law) that if A affects B, then B affects A.

These two differences to me mean that Bohmian mechanics is not just a classical hidden variables theory of the type Einstein was hoping for, and it's not just the nonlocal interactions. It seems to me that Bohmian mechanics requires all the ontology of Many-Worlds in order to have a universal wave function that is a real, physical thing. To me, it seems to be adding something on top of Many-Worlds, which is the initial locations of particles.

I actually think that that might be just what Many-Worlds needs, an unambiguous definition of what a "world" is. Bohmian mechanics basically says that a world is determined by an initial specification of the locations of all the particles. It also gives a simple solution to the problem of the meaning of probability in Many-Words. You don't try to derive the probability from the wave function alone, you just assume that the particle locations are distributed according to the wave function via the Born rule.

 

[Demystifier]

I don't think that BM can be considered to be on par with the other interpretations of QM.

Why not? Other interpretations also have heuristic claims which cannot be proved rigorously. For instance, MWI has a notoriously hard problem of justifying the Born rule. Copengahen (at least the Bohr's version of it) claims that there is a borderline between micro and macro worlds, one obeying quantum and the other classical laws, which is also an unproved statement. Do you know some interpretation which does not rest on unproved statements?

 

[Demystifier]

@stevendaryl, I agree with you that BM can be thought of as a completion of the MWI idea.

Concerning the fact that particles do not influence the wave function which lives in the configuration space, note that this makes wave function similar to the Hamiltonian in classical mechanics. In this sense classical mechanics is a 2-level theory:
Hamiltonian -> particle trajectories
which can also be formulated as a 3-level theory
Hamiltonian -> principal function (the solution of Hamilton-Jacobi equation) -> particle trajectories
very similar to Bohmian mechanics which is also a 3-level theory:
Hamiltonian -> wave function (the solution of Schrodinger equation)-> particle trajectories

 

[rubi]

Why not? Other interpretations also have heuristic claims which cannot be proved rigorously. For instance, MWI has a notoriously hard problem of justifying the Born rule. Copengahen (at least the Bohr's version of it) claims that there is a borderline between micro and macro worlds, one obeying quantum and the other classical laws, which is also an unproved statement. Do you know some interpretation which does not rest on
unproved statements?

I agree that MWI also has serious problems. I don't agree that Copenhagen claims that anything obeys classical laws. No Copenhagen-style textbook ever makes this claim. Copenhagen says that in the case of a measurement, a different (but also non-classical) time evolution law applies and this is an axiom. No mathematical contradiction can arise from it. In BM, you will get a mathematical contradiction if the deterministic laws determine other probabilities than those that BM claims to predict. Only one set of probabilities can be correct, so they better agree. One may find the Copenhagen time evolution prescription unrealistic (in which case one should check out the consistent histories formulation), but at least, it is mathematically consistent and doesn't contain unproved statements (other than axioms of course). Problems only arise if the axioms contradict each other, which doesn't happen in Copenhagen, but might happen in BM.

 

[Demystifier]

and big names like Hagen Kleinert appear among the authors.

As I already explained
https://www.physicsforums.com/threa...l-issue-with-pilot-waves.893734/#post-5622848
he makes the same mistake as the other 99% I refer to in post #3.

 

[Demystifier]

Copenhagen says that in the case of a measurement, a different (but also non-classical) time evolution law applies and this is an axiom. No mathematical contradiction can arise from it.

There is no mathematical contradiction simply because the statement is mathematically so much vague that it is not even wrong. I mean, from that statement it is not clear at all what physical processes do and which physical processes don't count as measurements. By a vague statement it's very easy to avoid mathematical (or any other) contradiction.

 

[Demystifier]

Problems only arise if the axioms contradict each other, which doesn't happen in Copenhagen, but might happen in BM.

Can you name two axioms in BM which potentially might contradict each other? I have no idea what those could be.
On the level of rigorous mathematics, it is proved rigorously that the axiom
$$\rho(x,t_0)=|\psi(x,t_0)|^2$$
for an arbitrary initial ##t_0##, implies the theorem
$$\rho(x,t)=|\psi(x,t)|^2$$
for all ##t##. It is also quite clear that the axiom above does not contradict any of the other axioms. The only issue is whether that axiom is independent on other axioms, or can be derived from the other axioms. There are various arguments that it can be derived, but these arguments are not absolutely rigorous proofs.

 

[rubi]

As I already explained
https://www.physicsforums.com/threa...l-issue-with-pilot-waves.893734/#post-5622848
he makes the same mistake as the other 99% I refer to in post #3.

Well, I find Arnold's objection very reasonable. In a universe with only one hydrogen atom, there is nothing that could perturb the atom, so if Kleinert is right, then there seem to be systems with non-matching predictions between QM and BM.

There is no mathematical contradiction simply because the statement is mathematically so much vague that it is not even wrong. I mean, from that statement it is not clear at all what physical processes do and which physical processes don't count as measurements. By a vague statement it's very easy to avoid mathematical (or any other) contradiction.

But the situation is still much different. The axioms of Copenhagen are mathematically consistent. In order to match a mathematical theory to experiment, one of course always needs to specify how it relates to reality. In the case of Copenhagen, one needs to specify what a measurement is. However, this concerns only the physical side of the theory. For BM, it is already possible for the mathematical side of the theory to be inconsistent, i.e. proving a contradiction.

the axiom
$$\rho(x,t_0)=|\psi(x,t_0)|^2$$

It is also quite clear that the axiom above does not contradict any of the other axioms. The only issue is whether that axiom is independent on other axioms, or can be derived from the other axioms. There are various arguments that it can be derived, but these arguments are not absolutely rigorous proofs.

No, those aren't the only two possibilities. There is also a third possibility: An initial distribution can be derived from the theory (without using the above axiom), but it is not the one specified by the above axiom. In that case, the above axiom would contradict the remaining axioms and the theory would be mathematically inconsistent. If it is in principle possible to derive the above axiom, then it is also in principle possible that a different initial distribution could be derived.

 

[Demystifier]

Well, I find Arnold's objection very reasonable. In a universe with only one hydrogen atom, there is nothing that could perturb the atom, so if Kleinert is right, then there seem to be systems with non-matching predictions between QM and BM.

If by predictions you mean measurable predictions, then your statement is nonsense because QM makes no measurable predictions at all for a universe with only one hydrogen atom. In such a universe there are no measurements, and without measurements there are no measurable predictions.

Or perhaps by predictions you mean also non-measurable predictions? In that case I fully agree that BM and standard QM make different non-measurable predictions, but I don't see that as a problem for any of the two theories.

The axioms of Copenhagen are mathematically consistent.

Except those which are not mathematical at all, such as those that refer to measurement. Non-mathematical statements are mathematically neither consistent nor inconsistent.

In the case of Copenhagen, one needs to specify what a measurement is. However, this concerns only the physical side of the theory. For BM, it is already possible for the mathematical side of the theory to be inconsistent, i.e. proving a contradiction.

By Popper, a potential to prove or disprove a statement is a virtue, not a drawback.

There is also a third possibility: An initial distribution can be derived from the theory (without using the above axiom), but it is not the one specified by the above axiom. In that case, the above axiom would contradict the remaining axioms and the theory would be mathematically inconsistent. If it is in principle possible to derive the above axiom, then it is also in principle possible that a different initial distribution could be derived.

If I understood you correctly, you suggest that we should better use vague theories which cannot be disproved even in principle, because a mathematically precise theory can always be subject to a rigorous analysis, and rigorous analysis always offers a possibility to disprove the theory. Is that what you are suggesting? If it is, then your view is rather anti-Popperian and anti-scientific.

One additional comment. It is known that quantum field theory has mathematical contradictions due to the infinite number of degrees of freedom in the UV and IR limits. On the other hand, classical Newtonian mechanics does not have such contradictions. Should we then reject quantum field theory and use Newtonian mechanics instead?

 

[rubi]

If by predictions you mean measurable predictions, then your statement is nonsense because QM makes no measurable predictions at all for a universe with only one hydrogen atom. In such a universe there are no measurements, and without measurements there are no measurable predictions.

Or perhaps by predictions you mean also non-measurable predictions? In that case I fully agree that BM and standard QM make different non-measurable predictions, but I don't see that as a problem for any of the two theories.

Well, it's already an interesting result if the theories aren't mathematically equivalent, because that's what lots of people believe.

Except those which are not mathematical at all, such as those that refer to measurement. Non-mathematical statements are mathematically neither consistent nor inconsistent.

You can also formulate the axioms without mentioning physics at all. For example, you could say: Let ##(P_{t_i})_{t_i}## be a set of prejectors indexed by real values ##t_i##. The evolution law of a state ##[\Psi]## is given by ##[\Psi(t)]= [U(t,t_n)P_{t_n}U(t_n,t_{n-1})\cdots\Psi]##. Of course, it would be pointless to do this, but it is in principle possible.

By Popper, a potential to prove or disprove a statement is a virtue, not a drawback.

Popper's principle concerns experimental falsification of otherwise mathematically consistent physical theories. A mathematically inconsistent physical theory is already falsified before any experiment is made. There is no advantage of having a potential mathematical inconsistency in a physical theory.

If I understood you correctly, you suggest that we should better use vague theories which cannot be disproved even in principle, because a mathematically precise theory can always be subject to a rigorous analysis, and rigorous analysis always offers a possibility to disprove the theory. Is that what you are suggesting? If it is, then your view is rather anti-Popperian and anti-scientific.

No, that's not what I am suggesting. I don't like Copenhagen either and I too find it vague, which is why I don't usually advocate it. But you claimed that it contained unproved statements, which isn't true. I can assure you that I am a big fan of Popper. I was talking about the possibility of a mathematical inconsistency in BM, which is swept under the carpet by the BM community. The thread is about potential problems in BM and I think it is not helpful to ignore these problems by directing the attention to other interpretations. That would really be anti-scientific.

 

[Demystifier]

Well, it's already an interesting result if the theories aren't mathematically equivalent, because that's what lots of people believe.

Then let me repeat once again. BM and standard QM are mathematically not equivalent. But it does not mean that they are not observationally equivalent. Moreover, there is a theorem which proves that they are observationally equivalent. In principle it is not impossible that there is a gap in the proof, but then any critique of their observational equivalence should concentrate on finding the gap within that proof itself. Nobody so far have found such a gap within the proof. About 99% of the existing critiques of the equivalence do not refer to that proof at all, which makes such critiques worthless.

I was talking about the possibility of a mathematical inconsistency in BM, which is swept under the carpet by the BM community.

As Godel has shown in his second theorem, not even the standard axioms for arithmetic of integer numbers can prove their own consistency. It is therefore illusory to expect that axioms of any relevant physical theory (including Bohmian mechanics and standard QM) could prove it's own consistency. But even 99% mathematicians are not worried by the fact that they can't prove consistency of their favored axioms. Why should Bohmians be worried then? In practice, for 99% mathematics and for 99.99% physics, it is enough that the axioms look consistent intuitively. And so far, you haven't presented any heuristic argument (let alone proof) that some of the axioms for BM seem inconsistent.

I was talking about the possibility of a mathematical inconsistency in BM, which is swept under the carpet by the BM community.

As I said, nobody so far found any evidence (let alone proof) for such inconsistency. Concerning the proof of consistency, can you prove that standard QM is consistent? If you think you can, then you probably contradict the second Godel theorem.