# A An argument against Bohmian mechanics?

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1. Dec 21, 2016

### 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.

2. Dec 21, 2016

### StevieTNZ

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.

3. Dec 22, 2016

### 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.

4. Dec 22, 2016

### 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.

5. Dec 22, 2016

### A. Neumaier

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.

Last edited: Dec 22, 2016
6. Dec 22, 2016

### Demystifier

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

7. Dec 22, 2016

### A. Neumaier

Well, prove your claim by meeting my objection!

8. Dec 22, 2016

### Demystifier

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

9. Dec 22, 2016

### A. Neumaier

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!

10. Dec 22, 2016

### Demystifier

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.

11. Dec 22, 2016

### 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!

12. Dec 22, 2016

### Demystifier

Last edited: Dec 22, 2016
13. Dec 22, 2016

### Demystifier

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.

14. Dec 22, 2016

### A. Neumaier

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

15. Dec 22, 2016

### Demystifier

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
$$\langle \psi |x(t_1)x(t_2)| \psi\rangle$$

16. Dec 22, 2016

### A. Neumaier

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.

17. Dec 23, 2016

### 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.

18. Dec 23, 2016

### rubi

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.

19. Dec 23, 2016

### Demystifier

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.

20. Dec 23, 2016

### rubi

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.

21. Dec 23, 2016

### Demystifier

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.

22. Dec 23, 2016

### rubi

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.

23. Dec 23, 2016

### Demystifier

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.

24. Dec 23, 2016

### rubi

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.

25. Dec 23, 2016

### stevendaryl

Staff Emeritus
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, $\vec{v} = \frac{\hbar}{m}Im(log(\psi))$, 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" $V_Q = \frac{-\hbar^2}{2m} \frac{\nabla^2 |\psi|}{|\psi|}$). 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.