Consistency of Bohmian mechanics

[Thors10]

$E_t$ is the time evolution of the Bohmian hidden variables (from $t_0=0$ to $t$). In my example it is given by $E_t(x)=x\sqrt{1+t^2}$, so $E_t^{-1}(x) = \frac{x}{\sqrt{1+t^2}}$ and given $x(t)$, it just gives us $x(0)$. So if we learn something at time $t$, we compute what the initial conditions must have been through $E_t^{-1}$ and update the probability distribution on the initial conditions accordingly. In my example, $t$ was just $0$, so $E_0^{-1}(x) = x$ and $\chi_{E_0^{-1}(\mathbb{R}_+)}(x) = \chi_{\mathbb{R}_+}(x) = \theta(x)$.

 

[Thors10]

Yes, of course in standard QM we have to update the wave function due to collapse. The problem is that the updated Bohmian probability distribution of the full system is not equal to the collapsed and evolved QM wave function at later time. In this harmonic oscillator example, we would have to collapse the wave function to some function of $x$ times the delta distribution $\delta(y-y_0)$ and the Hamiltonian I wrote won't evolve this into the updated Bohmian distribution, because in BM the wave function of the full system evolves unitarily and the updating just restricts the possible initial conditions of the Bohmian variables.

 

[Demystifier]

Yes, of course in standard QM we have to update the wave function due to collapse. The problem is that the updated Bohmian probability distribution of the full system is not equal to the collapsed and evolved QM wave function at later time.

There are many texts in the literature that show that they are equal. It would help if you could take some of those texts and say what, in your opinion, is exactly wrong in this text.

In this harmonic oscillator example, we would have to collapse the wave function to some function of $x$ times the delta distribution $\delta(y-y_0)$ and the Hamiltonian I wrote won't evolve this into the updated Bohmian distribution, because in BM the wave function of the full system evolves unitarily and the updating just restricts the possible initial conditions of the Bohmian variables.

I don't quite follow your arguments, but note that the Hamiltonian you wrote is not an interaction that may serve the purpose of measurement. It's not that any interaction counts as measurement. A measurement is an interaction that splits the wave function into separated branches as in the drawing in my first post on this thread.

 

[Thors10]

What is conjectured in the literature is that BM reproduces the wave function collapse on the level of a subsystem. I'm not convinced by the arguments, because the calculations are too simplistic, but anyway, that's not what I care about. My question concerns the collapse of the full system's wave function, not the subsystem. I don't think this is discussed anywhere in the literature, but I would of course be interested if you could point me somewhere.

Well, as Elias1960 said, your drawing is misleading. One can achieve the split you want by turning the interaction on and off in some finite time interval. Nevertheless, the Bohmian updating won't agree with the quantum collapse. It's easy to see that this can only be true if the pointer variable commutes with the Hamiltonian and this is never the case, because then the pointer observable would be conserved. A measurement device with a constant pointer is broken.

 

[Elias1960]

Well, I don't care about subsystems. My issue concerns the full system and I'm asking whether BM produces the same predictions for the full system as QM. ... In my example, the probability distribution after measurement would be $2\theta(x)\left|\psi(x,t)\right|^2$, which is not equal to $\left|\psi(x,t)\right|^2$, if we consider for the moment for illustrative purposes the possibility that the full system consists only of one particle.

That's funny, you claim not to care about subsystems but then give a wave function for the subsystem.

Once you consider a measurement in BM, the full system you have to consider is always the one which includes the measurement device too, so that it simply cannot be a one particle system.

 

[Thors10]

Elias1960, just pretend that $x$ is already a macroscopic pointer observable which can be read off without disturbances, not a particle. It's only for illustrative purposes, because nothing changes in the argument if you include more variables.

 

[Elias1960]

Elias1960, just pretend that $x$ is already a macroscopic pointer observable which can be read off without disturbances, not a particle. It's only for illustrative purposes, because nothing changes in the argument if you include more variables.

Whatever, you have to consider the full system, with measurement devices, if you want something following the Schroedinger equation. For this full system, the standard equivalence theorem holds. If you want to see how the collapse happens, you have to look at the subsystem during the measurement, and its wave function is defined by the full wave function (following the Schroedinger equation) and the trajectory of the measurement device. This leads, applied to a particular trajectory of the measurement device, to a particular collapse trajectory.
For me, it looks like you have not yet understood the whole picture.

 

[Thors10]

Elias1960: Even though I don't think it's necessary, I did this already in post 28, so we can move on to the real problem I'm interested in.

The problem I see is the following. There are two types of variables needed in BM:
1. Microscopic variables, which can only be measured by interaction with a measurement apparatus. BM claims to be able to derive an apparent collapse for the subsystem that describes these variables. Alright, I won't question this anymore, if we can then agree to move on to the problem I really care about.
2. Macroscopic pointer variables, which can be read off directly from the device. Gaining knowledge about them doesn't require interaction. Instead we can just learn the approximate value of these variables.

The problem is that gaining knowledge and updating the probability distribution for variables of the second kind in BM is not in agreement with the the collapse of the wave function in QM. The literature in BM doesn't discuss this problem at all as far as I can tell. If you disagree, please point me to a paper or book that treats this.

 

[Demystifier]

My question concerns the collapse of the full system's wave function, not the subsystem.

You are right that in BM there is no collapse of the full system, including the measuring apparatus. But that's not much different from QM, because the standard textbook QM usually does not contain a collapse of the full system. That's because in QM textbooks the measurement apparatus is usually not represented by a wave function at all. Bohr, for instance, argued that measurement apparatus is classical. This does not contradict existing experiments because the wave function of a measurement apparatus (or any other system with many effective degrees of freedom) cannot be determined experimentally. In this sense, QM and BM have the same measurable predictions. They have some differences on how they treat the full system, but these differences cannot be measured.

 

[Elias1960]

The problem is that gaining knowledge and updating the probability distribution for variables of the second kind in BM is not in agreement with the the collapse of the wave function in QM. The literature in BM doesn't discuss this problem at all as far as I can tell. If you disagree, please point me to a paper or book that treats this.

The global wave function remains uncollapsed. So, it contains all the empty worlds which we already know to be irrelevant given that we have direct access to the trajectories. Schroedinger's cat's wave function remains to be a superposition of the living and the dead cat. But what we see is the trajectory of the cat, not the wave function. The collapse is something we have only if we reduce the global wave function to an effective one for some subsystem, using the trajectory we see. That's all.

 

[Demystifier]

2. Macroscopic pointer variables, which can be read off directly from the device. Gaining knowledge about them doesn't require interaction.

That's wrong, gaining knowledge about them requires interaction too. But since they are macroscopic, effects of those interactions can be approximated with classical physics.

 

[Thors10]

Demystifier:
Right, standard QM doesn't need to include the apparatus in the quantum system. But if we include the apparatus in the quantum system, standard textbook QM does need the collapse, because decoherence only gives us density matrix containing the incoherent mixture of all possible resulting wave functions. We need to collapse the full systems wave function to obtain a definite result. That's what the measurement problem is about. On the other hand, BM needs to update the probability distribution, because probabilities are supposed to be only due to ignorance. It is critical that these different processes of updating are in agreement, because otherwise they will lead to different predictions.

Elias1960:
The global wave function in BM remains uncollapsed, right. But nevertheless, the probability distribution must be updated, because probabilities are only due to ignorance in BM. If we gain knowledge, our ignorance reduces and the probability distribution must reflect that.

Demystifier:
What does the pointer observable need to interact with during the gaining of knowledge about it?

 

[Elias1960]

Elias1960:
The global wave function in BM remains uncollapsed, right. But nevertheless, the probability distribution must be updated, because probabilities are only due to ignorance in BM. If we gain knowledge, our ignorance reduces and the probability distribution must reflect that.

But it has to be updated because we see a measurement result. We see it, it is, therefore, not in the quantum part, it is described not by the wave function, but by the trajectory. There is no point in using a wave function once the real information about the trajectory is available. The part which will be described by a wave function is what remains.

 

[Thors10]

What is your point? Do you disagree that we need to update our probability distribution once we learned something about the system? I'm just talking about standard Bayesian updating. BM tells us that our particles are initially distributed according to $\left|\psi(\mathbf{X},0)\right|^2$. After gaining knowledge, this distribution can no longer be maintained. The new probabilities will be given by $P'(X)=P(X|A)$ and the probability distribution function of $P'$ will be given by $\frac{1}{P(A)}\chi_A(\mathbf{X})\left|\psi(\mathbf{X},0)\right|^2$.

My point is that the time evolution of this new probability density disagrees with the time evolved probability we get from QM and thus BM and QM make different predictions.

In BM, the time evolution of the updated probability density is given by $\frac{1}{P(A)}\chi_A(E_t^{-1}(\mathbf{X}))\left|\psi(E_t^{-1}(\mathbf{X}),0)\right|^2 = \frac{1}{P(A)}\chi_{E_t(A)}(\mathbf{X})\left|\psi(\mathbf{X},t)\right|^2$.

In QM, the time evolution of the updated probability density is given by $\frac{1}{P(A)}\left|(U(t)\chi_A({-})\psi({-},0))(\mathbf{X})\right|^2$ and this is incompatible with the Bohmian formula.

If BM and QM are supposed to be equivalent, then these two formulas must necessarily be equal. There just isn't any other way. So the way I see it after this discussion so far is that BM and QM are really inequivalent theories (which may occasionally make similar predictions).

 

[WWGD]

Just don't malign Bohmian Rhpsody.

 

[Thors10]

Just to make it completely clear:

I consider a system that starts with an initial wave function $\psi(\mathbf{X},0)$. Then I perform a position measurement at time $t=0$ and find the system in $\mathbf{X} \in A$. Then I evolve the system to time $t$. I look at the evolution of the initial probability density of this process in both BM and QM.

In BM, this works as follows:
$\left|\psi(\mathbf{X},0)\right|^2 \rightarrow \frac{1}{P(A)}\chi_{A}(\mathbf{X})\left|\psi(\mathbf{X},0)\right|^2 \rightarrow \frac{1}{P(A)}\chi_{E_t(A)}(\mathbf{X})\left|\psi(\mathbf{X},t)\right|^2$

Orthodox QM has the following evolution:
$\left|\psi(\mathbf{X},0)\right|^2 \rightarrow \frac{1}{P(A)}\left|\chi_{A}(\mathbf{X})\psi(\mathbf{X},0)\right|^2 \rightarrow \frac{1}{P(A)}\left|\left(U(t)\chi_{A}({-})\psi({-},0)\right)(\mathbf{X})\right|^2$

The math is correct and it's also the correct application of the BM and QM formalisms. So the question to BM is whether these formulas agree. But it's clear that they don't in general, because there are counterexamples. Hence, BM and QM are not equivalent at the level of full systems.

 

[Demystifier]

Demystifier:
Right, standard QM doesn't need to include the apparatus in the quantum system. But if we include the apparatus in the quantum system, standard textbook QM does need the collapse, because decoherence only gives us density matrix containing the incoherent mixture of all possible resulting wave functions. We need to collapse the full systems wave function to obtain a definite result. That's what the measurement problem is about. On the other hand, BM needs to update the probability distribution, because probabilities are supposed to be only due to ignorance.

So far I agree.

It is critical that these different processes of updating are in agreement, because otherwise they will lead to different predictions.

I agree that the two approaches make different "predictions" here, but I do not think that this difference can be measured in practice. Or if you think that it can be measured in practice, can you propose a specific experiment how that could be done?

Demystifier:
What does the pointer observable need to interact with during the gaining of knowledge about it?

Maybe I am missing the true motivation for this question, but obviously it must interact with the measured system. A nontrivial question is precisely how they should interact. For a model of an appropriate interaction see http://de.arxiv.org/abs/1406.5535 Sec. 6.1.

 

[Demystifier]

Hence, BM and QM are not equivalent at the level of full systems.

I agree, but the claim is that this difference cannot be measured in practice. In principle the difference could be seen if one could prepare a true macroscopic Schrödinger cat state, but nobody has done it so far. In existing experiments it is possible to prepare a mesoscopic Schrödinger "cat", which shows that there is no collapse at the mesoscopic level, which can be taken as an experimental indication that the Bohmian interpretation could be closer to truth than the collapse interpretation. But it is consistent with a collapse interpretation too if one claims that the collapse only happens at the macroscopic level, and not on the mesoscopic one.

 

[Thors10]

Well, in order for the different predictions to be measured in practice, one would need an actual prediction of BM in the frist place. However, I don't think any experimentally relevant prediction of BM has ever been computed at all. There is only the claim that BM will make the same predictions as QM and therefore we should just compute the predictions in QM. But that's not what needs to be done if we want to test BM. We need a full BM computation that we could compare to experiment. Otherwise it's just a test of QM, not of BM.
Regarding the pointer observable: I said that these observables can be read off directly from the device, unlike the observables of the particles. Of course the pointer variable interacts with the particle, but in order to gain knowledge about it, I don't need to include yet another interaction term with some additional system.

You said: "That's wrong, gaining knowledge about them requires interaction too. But since they are macroscopic, effects of those interactions can be approximated with classical physics."

I asked you what these classical interactions would be, because certainly you can't mean the quantum interactions with the particle? They are not classical at all.

My point is just: In order to measure the position of a particle in BM, I'm told that I can't just take its Bohmian trajectory. Instead I must couple the particle to some measurement device and instead measure the position of the pointer variable. So the pointer variable is of a different kind: I don't need to couple it to yet another bigger system in order to gain knowledge about it. I can just take the actual value of its Bohmian trajectory.

 

[Thors10]

"I agree, but the claim is that this difference cannot be measured in practice."
Well that's the problem I'm having. The BM community seems to shy away from making predictions with their theory. I'm trying to understand the (in)equivalence by looking at some actual calculations, but I'm just told that they are not sufficient. Then again, nobody seems to have done a sufficient calculation at all, so I don't understand the confidence in the claim. I'm just interested in some quantitative computations in BM, but I only ever get loose arguments. And those arguments never suffice to get some quantitative results out of them that could in principle be compared to experiments.

 

[Demystifier]

"I agree, but the claim is that this difference cannot be measured in practice."
Well that's the problem I'm having. The BM community seems to shy away from making predictions with their theory. I'm trying to understand the (in)equivalence by looking at some actual calculations, but I'm just told that they are not sufficient. Then again, nobody seems to have done a sufficient calculation at all, so I don't understand the confidence in the claim. I'm just interested in some quantitative computations in BM, but I only ever get loose arguments. And those arguments never suffice to get some quantitative results out of them that could in principle be compared to experiments.

Well, some authors do try to make new measurable predictions out of BM, but in my opinion such attempts are misguided. See my "Bohmian mechanics for instrumentalists" (linked in my signature below), Sec. 4.4.

 

[Thors10]

How could such predictions be misguided? Either BM and QM can be shown to be equivalent. This seems not to be the case. Or we need to understand their difference quantitatively, in order to test which of them is correct. Without quantitative computations, BM can't be said to be compatible with experiments, yet.

I have looked at the section of that paper, but I still don't understand why we should allow BM to get away with not making quantitative predictions? Every physical theory must do that at some point. It's not a "trap" as you call it in that paper. It's just how science works.

 

[Demystifier]

How could such predictions be misguided? Either BM and QM can be shown to be equivalent. This seems not to be the case. Or we need to understand their difference quantitatively, in order to test which of them is correct. Without quantitative computations, BM can't be said to be compatible with experiments, yet.

I have looked at the section of that paper, but I still don't understand why we should allow BM to get away with not making quantitative predictions? Every physical theory must do that at some point. It's not a "trap" as you call it in that paper. It's just how science works.

Obviously you have not understood my argument in the paper, so let me rephrase it. If practically measurable differences between two theories exist, then they should be pointed out. But if they do not exist, then honest scientists should not pretend that they exist just for the sake of looking "more scientific". I argue in the paper that they do not exist, and that authors who claim the opposite either misunderstand BM or pretend that they exist for the sake of looking more scientific. If you still don't understand it, then I don't know how to put it differently.

But note that latter in the paper I do make a new generic measurable prediction out of BM, but on an entirely different level. It's in Sec. 5.2.

 

[Thors10]

Demystifier:
"I argue in the paper that they do not exist"
Well, but I don't see any quantitative argument in the paper. You make a lot of approximations and you don't quantify how good these approximations are. You would have to work in an actual model to calculate such deviations. Only then can we test whether BM gives the same predictions as QM up to the measurement uncertainty of our state of the art experiments. Also you don't study how quickly the deviations will evolve. Maybe they are small at first but will explode after a short time interval. All of this must be checked. Just claiming it, isn't enough.

Let me give an example:
If I stack two spheres exactly on top of each other, then classical mechanics tells us that they will stay in this state forever. Now if I fail to find the right spot and make a small deviation, then you would argue that these deviations are small and because of this I should still expect the spheres to remain almost motionless. But the truth is that this situation is unstable and the deviations will quickly explode and the system will collapse. One can study this instability using math and it's very important in order to make the correct predictions. If we don't know quantitatively how the deviations will manifest themselves, the theory is not predictive at all.

 

[Demystifier]

Demystifier:
"I argue in the paper that they do not exist"
Well, but I don't see any quantitative argument in the paper. You make a lot of approximations and you don't quantify how good these approximations are. You would have to work in an actual model to calculate such deviations. Only then can we test whether BM gives the same predictions as QM up to the measurement uncertainty of our state of the art experiments. Also you don't study how quickly the deviations will evolve. Maybe they are small at first but will explode after a short time interval. All of this must be checked. Just claiming it, isn't enough.

In principle you are right. But I have seen a lot of papers in which calculations of that sort have been done explicitly, so I think I can say I have a good intuition about which effects can be significant and which can't. It's not a proof that I am right, but all physicists have some intuition about some effects that makes them fairly confident in ignoring some effects due to being negligible. In practice, it's impossible to check everything in a single paper. Theoretical physics is, among other things, an art of making approximations based on experience. The point of this paper is to convey to other physicists my intuition about this stuff, not to make proofs. The proofs, or at least more quantitative arguments, can be found in references I cited. In this paper I am trying to explain the big picture, not the details. I am showing the woods, not the trees. Anyone who wants to see more details can read the cited references and make some detailed calculations by himself.

 

[Thors10]

Alright, then thanks for the discussion, I think it enhanced my understanding of BM quite a bit. Initially I thought that BM is really an exact interpretation of QM and gives the exact same results and I was just missing some basic argument. Now that I understand the difference, I think one should see this inequivalence as a chance, because it makes the question of interpretations amenable to experimental tests rather than a matter of opinion. I think the most important task for the BM community is to come up with some general methods of quantifying these deviations, so the question, which interpretation is right, can ultimately be answered by experiments, just like with Bell's theorem, which made some deep interpretational questions accessible to experimental physics. This could be really exciting.

 

[Demystifier]

Alright, then thanks for the discussion, I think it enhanced my understanding of BM quite a bit. Initially I thought that BM is really an exact interpretation of QM and gives the exact same results and I was just missing some basic argument. Now that I understand the difference, I think one should see this inequivalence as a chance, because it makes the question of interpretations amenable to experimental tests rather than a matter of opinion. I think the most important task for the BM community is to come up with some general methods of quantifying these deviations, so the question, which interpretation is right, can ultimately be answered by experiments, just like with Bell's theorem, which made some deep interpretational questions accessible to experimental physics. This could be really exciting.

In principle, I agree. In fact, in my younger days I was trying myself to make such a measurable distinction, see http://de.arxiv.org/abs/quant-ph/0406173 . But later, with more experience, I realized that my attempt (as well as attempts of many others) was too naive.

 

[Thors10]

Here's another question for you: If you think one can do QM without intermediate collapse, then how do you calculate probabilities such as $P(\mathbf{X}(t_1) \in A_1 \wedge \mathbf{X}(t_2) \in A_2)$? In orthodox QM, you would just calculate $\int \left|(\chi_{A_2}U(t_2-t_1)\chi_{A_1}U(t_1)\psi)(\mathbf{X})\right|^2 \mathrm{d}\mathbf{X}$, but you can't do that without collapse, because the Heisenberg position observables $\hat{x}(t)$ don't commute for different $t$ (not even if they are pointer positions), so one can't perform the calculation at the end of the time evolution. One has to insert the $\chi$'s at intermediate times.

 

[Elias1960]

Here's another question for you: If you think one can do QM without intermediate collapse, then how do you calculate probabilities such as $P(\mathbf{X}(t_1) \in A_1 \wedge \mathbf{X}(t_2) \in A_2)$? In orthodox QM, you would just calculate $\int \left|(\chi_{A_2}U(t_2-t_1)\chi_{A_1}U(t_1)\psi)(\mathbf{X})\right|^2 \mathrm{d}\mathbf{X}$, but you can't do that without collapse, because the Heisenberg position observables $\hat{x}(t)$ don't commute for different $t$ (not even if they are pointer positions), so one can't perform the calculation at the end of the time evolution. One has to insert the $\chi$'s at intermediate times.

Counterquestion: How do you measure it? To compare your probability with observation, you have to have information about what was at $t_1$ as well as at $t_2$. How is this done in QM? Very simple, you measure $A_1$ at $t_1$, and care about that the measurement result is preserved. So, you start with
$\psi_1(a_1) \psi_2(a_2)\psi(q,0)$. Independent variables defining the states of the two measurement devices and the state of the system, and independent initial wave functions. And physics which makes the interaction between them zero except for the time of the measurements. At $t_1$ this will become a superposition
$$\sum_i c_i \psi^i_1(a_1,t_1) \psi_2(a_2,t_1)\psi_i(q,t_1).$$
After this, the first measurement result will be preserved, thus, no more interaction with all the other things, and trivial evolution of $a_1$ itself. Thus, the future evolution of the $\psi^i_1(a_1,t)$ will be trivial after $t_1$.
At $t_2$ this will become a superposition
$$\sum_i c_i \psi^i_1(a_1,t_1) \sum_j c_{ij} \psi^j_2(a_2,t_2)\psi_{ij}(q,t_2).$$

This is all yet quantum evolution without collapse, without any $\chi$ inserted. Then comes the final measurement of the positions $a_1, a_2, q$ which gives you $\rho(a_1, a_2, q, t_2)$. It measures positions of different objects, they commute, no problem.

 

[Demystifier]

As far as I can see, whatever is described here as a difference between BM and QM appears as well in QM itself if we use different cuts.

I agree.